The 16th century is old enough that some people groups went through name changes, and many of the names you are familiar with are not on here. This map also prioritizes the endonym (what they call themselves) name over any exonyms (what other people call them). Some exonyms are just anglicisations of the endonym (or another European language), or sometimes they are direct translations or the native name, or names from another tribe.
Apsáalooke means "crow" in the Apsáalooke language, so most people call them Crow.
Additionally, writing these names in the Latin alphabet is not always done the same.
I do find it hard that you wouldn't recognize more than one name, though. I'm no scholar, but I counted at least 140 that I recognized, and I'm sure I would recognize more of them under other names.
If there's a tribe who's name you know that isn't on here, it's probably because it was a less popular exonym (like souix, for lakota), or they didn't exist yet (like the Seminoles), or it refers to more than one tribe (like iroquois).
Maybe I’m just seeing what I want to see, but these areas seem to more closely represent natural regional divisions—the desert area has one group, the hills another, etc. I imagine it made your community feel pretty cohesive.
Keep in mind that they while they did have tribal territory and such these aren't borders per se, so for quite a lot of this map their "territory" follows those regions because it's convenient for the map maker and they don't want to leave gaps!
They actually are on there. Comanches were originally a subgroup of the shoshoni, who came down from wyoming area after mastering the horse. They call themselves the Nʉmʉnʉʉ, which you can see on the map at the eastern edge of the green shoshoni lands.