Like the whole point of this little tribe
Horses are very sacred, a gift from their deity. Eating horseflesh (as well as dogs) is strictly forbidden. Every new member (at birth, full assimilation, whatever) gets a horse. The only answer to killing a horse from the tribe is overwhelming destruction. A tribe of hunter gatherers stupid enough to do so will have the men killed, camp looted and women and children taken back.
A new member (new baby born in the tribe, young child brought into the tribe, non tribe adult who earns their place in the tribe) will get a colored band made from the hair of the mane of their fathers best horse. If the person is not born in the tribe before earning their band another male tribe member will volunteer their horses mane.
Mares are more highly regarded than geldings, and more so than stallions. (At this point sexually mature stallions are still very aggressive and difficult)
They keep a handful of stallions for breeding. Every five years or so they look through the new stallions coming in and switch out older ones. The older stallions and those that don't make the cut are gelded and used regularly.
Pure white horses are rare, sacred and reserved for the speakers (tribe religious leaders and head midwife (If female))
They are this shape and size but not always these colors. There's more creative leeway as long as it's sensible. Stripes? Sure. Paints, appoolosa esque, solid, whatever. Pink polka dots on blue coat? No. Just no
Each horse has a strong personality, often very temperamental. They are very headstrong and quite attached to whoever they get attached to.
Each person has at least two. The owner chooses when and to which stallion their mares are bred.
When anyone is banded (where a new person is given the horse hair band that signifies they joined the tribe), all the mares and geldings are gathered and the newly banded person is brought forward. Out of the entire population one horse will stay with the person.