The Otsutsuki don't have the Sharingan, so that means that the Sharingan is an alien mutation, a unique genetic hybrid dojutsu unique to humans that originated from a combination of human and Otsutsuki DNA.
It's interesting that the Byakugan genetically transferred to Kaguya's descendants unaltered with its usual properties, yet aside from Hagoromo, Kaguya's descendants didn't directly inherit the Rinnegan, but rather an entirely different hybrid dojutsu. While the Sharingan can eventually upgrade into the Rinnegan under extremely unique and rare circumstances, and only for Indra's reincarnations, for most Uchiiha, the Eternal Mangekyou Sharingan was as far as they could go in leveling up their Sharingan.
Hagoromo was the first human to possess the Sharingan (naturally, being the son of Kaguya as well as the father of Indra Otsutsuki, the ancestor of all Uchiha) as well as the Rinnegan. Since Hagoromo's Otsutsuki blood was far less diluted than that of his descendants several millennia later, the descendants we see in the era when the Naruto series takes place, it should be noted that the rules and conditions for Hagoromo's dojutsu vary and differ a bit due to having a higher concentration of Otsutsuki blood.
The Sharingan is not a given ability, since merely being born of the Uchiha bloodline isn't enough to possess a Sharingan. They have to be awakened by a specific emotional response (with the Mangekyou Sharingan demanding the most specific set of circumstances of all).
The Sharingan grants its user many different abilities, with some of its basic functions including the ability to watch and then instantly copy others' movements and jutsu, to detect and predict the flow of an opponent's movements so they can easily dodge attacks, and the ability to cast genjutsu. It grants them the ability to see far, to see through walls, to see chakra (and the different colors of chakra), and even to use the Sharingan as a microscope and view things on a cellular level. It can read minds, search memories, and even erase memories, among other things.
The Itachi novels once mentioned that the Sharingan tints one's vision with a reddish hue, though how accurate or consistent this is is unclear, since it seems that there are different vision "modes" to the Sharingan, so it seems like it might be possible for a user to switch between modes, so they can either focus on predicting their enemy's movements or on looking at other people's chakra, or reading someone's mind, like Sasuke does when he invades Naruto's mind when he first meets Kurama.
So if the Sharingan actually tints a user's vision red, I don't think that the Sharingan's vision is always tainted a reddish hue (though it certainly could be and we just aren't ever shown the Uchiha's visual POV in the anime). Plus that's a novel and certain details from the novel are not necessarily always the most credible, even if I'd consider the Itachi novels mostly canon overall.
Although I think that when an Uchiha activates the Sharingan, they intentionally have to look for chakra flows to see them. Otherwise, it would be weird for characters like Sasuke, Sarada, and Madara to not register the change in their eyes right away when they awakened their Sharingan if their vision was so different; they should have noticed their newfound ability to suddenly see chakra or the red tint to their vision. So that tells me that these abilities require a bit of effort, and the way the Sharingan's activation changes vision isn't necessarily noticeable to its user.
The Sharingan is aptly nicknamed "the eyes that reflect the heart", because as much as they are a testament to the power of an Uchiha, they are a testament to the emotions of an Uchiha, a visual symbol of the user's emotions. Sharingan reacts to its users' negative emotions, and it is those negative emotions that cause it to awaken (and sometimes to activate), and the eyes that manifest as a result are an indication of the intensity of their emotional state, a symbol of past suffering and loss, or indicative of the user having experienced a traumatic incident in their past.
The Sharingan appears to be a dominant genetic trait, as a lot of the Uchiha we see have it, even half-Uchiha like Izumi and Sarada. Although genin Sasuke claims the Sharingan is rare among the Uchiha and unique only to a small number of the Uchiha clan, I have to question the validity of this statement, since it seemingly doesn't reflect what we're shown, though admittedly the Naruto series shows very little of the overall Uchiha populace aside from a few select members. Aside from a few minor characters, most of the Uchiha were the standout and exceptional of their world, not the average Uchiha. Though even amongst the background or minor Uchiha characters, a fair portion of them seemed to have Sharingan.
I'm skeptical of the claim of the Sharingan being "rare" among the Uchiha, since considering the violent and war-torn society that most Uchiha grew up in, it seems like there was plenty of emotionally traumatic and stressful scenarios to awaken the Sharingan, so presumably Uchiha would be awakening their Sharingan all the time. Especially since virtually all the main Uchiha characters awaken their Sharingan before they even hit adulthood:
Indra - Age 12
Madara - Around Age 15
Obito - Age 12 or 13
Itachi - Age 8
Izumi - Age 5-6
Sasuke - Age 7
Sarada - Age 5 (or Age 11 if you discount the anime)
Naruto is very much a "show don't tell" series, meaning that not everything a character says can be taken at face value, especially depending on the limitations of their knowledge. It's hard to guess how many Uchiha clan members that 7-year-old Sasuke was aware possessed the Sharingan. Did Sasuke know everyone in his clan who had a Sharingan? Did he have an accurate numerical guesstimate of the Sharingan's rarity? Did the Uchiha teach him a percentage of clan members who had the Sharingan versus those who didn't? Or could it be that little kid Sasuke had limited awareness of everyone who had a Sharingan in his clan and he made the assumption it was rare from his limited perspective? Maybe he spent more time around the civilian Uchiha who never saw action and therefore had never gained the Sharingan. Maybe not all Uchiha flaunt that they have the Sharingan and tend to keep it to themselves, so kid Sasuke is unaware of how many Uchiha have the Sharingan and assumes it's rare even though it actually is pretty common for an Uchiha. After all, of all the known Uchiha listed by name in the series, 74% of them have Sharingan, so I'm a bit skeptical that the Sharingan is all that rare among the Uchiha when nearly three-fourths of the named clan members in the series have it.
I also question the validity of Sasuke's statement on the Sharingan being rare because Sasuke isn't exactly the most knowledgeable person in the series. The Sharingan doesn't appear to be openly discussed with children much, since neither 8-year-old Itachi nor 7-year-old Sasuke knew the conditions for awakening the Sharingan, judging from his questions about it. And during Sasuke and Fugaku's conversation about the Sharingan, Fugaku remarks that, "it's too early for you". So I'm guessing discussion about the Sharingan is generally reserved more for teens and adults, not young children, so kid Sasuke wouldn't be well-informed about the Sharingan since his clan was killed off before he had the chance to learn his clan's lore.
Besides, Sasuke may not yet be aware that the regular Sharingan is awakened through emotional trauma and distress. And if Sasuke doesn't have any clue how the Sharingan awakens, he probably assumes it's through training or skill or power levels or some such conditions, not a power awakened by negative emotions and traumatic distress. Which means that the "rarity" of the Sharingan awakening that Sasuke postulates might be due to the rarity of the circumstances causing the awakening, not due to the latent potential in the Uchiha clan members being unique only for a small number of the Uchiha clan. Particularly if Sasuke hung around a lot of civilian Uchiha (and he probably would, being a civilian himself), they wouldn't be as likely to have the Sharingan as much because they had lived more peaceful lives and were less likely to have ever been under the kind of duress it usually takes to awaken a Sharingan, which would be a lot more common among the police force or ninja warriors of the clan. Which could mean that the "rarity" of the Sharingan awakening only among a small number of his clan has less to do with the clan's genetic potential being limited to only a few members, and is actually due to the rarity of the circumstances required to awaken the Sharingan.
And if Sasuke was clueless as to what triggered the regular Sharingan to awaken (and that does appear to be likely), he might mistakenly assume that it's the latent potential itself that is rare, being oblivious to the fact that it is actually the circumstances, not the potential, that is rare.
Anyway, Sasuke's information on this detail might be faulty, considering that Sasuke was limited in knowledge on other things, like never even having heard of Madara despite Madara being one of the village's co-founders, a famous member of the Uchiha clan, and there being a good chance that Madara is Sasuke's several times great-grandfather (evidently the history curriculum at the ninja academy is terrible or nonexistent, and it's pretty obvious that the shinobi world is quick to forget and doesn't put much effort into preserving history). Sasuke didn't know what a jinchuriki or tailed beast was until he was 17 years old despite having one on his team. Because Sasuke didn't have access to his clan's wealth of knowledge about the Sharingan, it would make sense for him to make an inaccurate deduction about the rarity of the Sharingan based off his limited knowledge.
Fans can draw whatever conclusions they wish to from Sasuke's statement, but considering the high percentage of Uchiha with Sharingan seen throughout the series (and not just the major characters, but the minor background Uchiha as well), I don't think the ability to awaken the Sharingan is rare. In fact, I personally assume all bloodline Uchiha possess the latent potential to awaken the Sharingan, it's just a matter of whether an Uchiha ever actually finds themselves in the circumstances and under high enough levels of emotional distress that trigger the Shairngan's awakening or not. Otherwise, we just coincidentally never got any information or news of any Uchiha who had to struggle with the unfairness of suffering great loss, grief, and guilt but not getting any power from it like the rest of his clanmates did, since all the Uchiha we saw who went through a traumatic experience got the Sharingan upgrade.
Anyway, this analyses is a compilation of my notes, thoughts, and observations on Sharingan lore and the rules of how the Sharingan works. I'm not including Sarada's Mangekyou awakening, since that plot point disregarded all the previously established canon rules of how the Sharingan works and frankly made no sense. Since Sarada's MS awakening doesn't apply to the rules of Sharingan lore, I will discuss it separately in its own chapter and omit it from consideration for the rest of the rules. I am also not taking Shin's Mangekyou Sharingan into account either, since he stands as another outlier to Sharingan lore. Since we have so little information on how Shin acquired the Mangekyou Sharingan and from whom, or how cloning works in the Naruto-verse, I am also going to discuss Shin separately in his own chapter.
For those unfamiliar with Sharingan slang, here's a glossary of terms for the Sharingan so readers don't get too confused with what level of Sharingan I'm referring to:
Tomoe/Magatama – The comma shapes that circle around the pupil of a Sharingan eye. A regular Sharingan will have either one, two, or three tomoe. Rinnegan can have several tomoe.
Phase I Sharingan – A Sharingan with one tomoe.
Phase II Sharingan – A Sharingan with two tomoe.
Phase III Sharingan – A Sharingan with three tomoe.
Mangekyou or MS – Abbreviation for Mangekyou Sharingan.
EMS – Abbreviation for the Eternal Mangkeyou Sharingan
Awaken – Just to clarify to prevent confusion later down the road, when I say "awaken the Sharingan", I'm going to use that to reference the first time an Uchiha activates and unlocks the Sharingan.
Activate – "Activate the Sharingan" is when an Uchiha switches on their Sharingan (that they have already previously awakened). I figured it might be helpful to distinguish how I'm using "awaken" and "activate" throughout the book just to avoid possible confusion, and I won't be using those words interchangeably so as to make sure the separate meanings stay clear.
Dojutsu – A dojutsu is a visual kekkei genkai, or an eye with a kekkei genkai power. Dojutsu is a broad category term, so it can refer to any kind of visual jutsu, including the Sharingan, Rinnegan, Tenseigen, Jogan, Ketsuryugan, etc.