Obito and Kakashi's simultaneous awakening of the Mangekyou Sharingan is a bizarre and unusual occurrence, one that might even puncture a gaping hole in Tobirama's scientific explanation of the biological reaction that causes an Uchiha's Sharingan to awaken.
If what Tobirama says about the Sharingan being triggered by a chemical released in the Uchiha's brain when they experience certain emotions is true, then theoretically a non-Uchiha like Kakashi should never be able to awaken the Mangekyou. So possibly this incident renders Tobirama's intel on the Sharingan as incorrect misinformation, not actually valid information on the Sharingan.
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Though truthfully I don't think Kakashi awakened Obito's Mangekyou Sharingan. Rather, I think it was Obito's emotional reaction to Rin's death that triggered the awakening in both eyes. And the awakening of Obito's Mangekyou caused a chain reaction that made his other eye, the one in Kakashi's possession, awaken in response, as a linked awakening.
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Therefore, it was Obito, not Kakashi, who awakened the Mangekyou, and Obito awakening the Mangekyou Sharingan in one eye triggered his other Mangekyou to respond the same way, thus why both eyes awakened at the exact same time.
Otherwise, if Kakashi awakened Obito's Mangekyou, it invalidates Tobirama's theory, since the Sharingan is supposed to awaken in response to a chemical reaction in an Uchiha's brain, as Uchiha are born with a unique chakra, the type of chakra that Kakashi wouldn't have.
If Kakashi's emotions were indeed what awakened the Sharingan, then we can assume that Tobirama's "scientific" information is incorrect, and that the chakra that possesses the capabilities to awaken the Sharingan is not unique to an Uchiha's physiology, but is present in all human beings.
However, the linked Sharingan awakening could possibly be made to fit with Tobirama's theory if one assumes that a pair of Sharingan eyes is inextricably linked (and this is shown to be true in Obito's case), so that whenever one eye is awakened to the Mangekyou level, it's counterpart must also upgrade at the same time, which is why Kakashi's Mangekyou triggers and awakens in response to its sibling eye, thus making them both into Mangekyou Sharingan at the same time.
Of course, Sasuke's Phase II Sharingan does perhaps disprove this theory, as he is the only user to upgrade his Sharingan unevenly, with one eye being Phase I and the other Phase II, so presumably for a regular Sharingan, the different phases can be awakened at different paces or one eye might lag behind a little before it catches up to the other (after all, after the Land of Waves mission, it didn't take Sasuke's Phase I Sharingan very long to fully upgrade to a Phase II Sharingan and become a matching pair again). So the Mangekyou level could work differently than the regular Sharingan level in this regard.
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Additionally, there were times when Kakashi could "link" to Obito's vision and see what was happening through Obito's eyes because they shared a Sharingan, meaning that Obito's Sharingan eyes had a shared connection. We could probably assume that's true for all Sharingan, and that a split pair shares a special connection, one that allows two separate users to see through another's vision, at least in some instances. However, a strong emotional bond and connection might be necessary for this linked vision, since Obito and Kakashi were close friends. And since Kakashi didn't randomly get images of what Obito was doing popping into his mind until he found out Obito was his friend and former teammate and had that emotional connection to him rekindled, it does seem that a conscious emotional connection and some friendly bond (or at the very least a formerly close friendship) is required for the linked vision to work. Though this is an underexplored bit of Sharingan lore, since sharing eyes doesn't happen often, so there isn't much observable data on this point. After all, most Uchiha wouldn't willingly relinquish their eyes or loan them to another except in a few rare special cases.
