The three spirits that possess the player characters are interchangeable, and all have a different effect on the protagonists.

Aside from the changes to gameplay, they also have a pronounced effect on the chosen character’s personality:

Wolf Spirit - Personality becomes barbaric.

Otter Spirit - Personality becomes wishy-washy.

Eagle Spirit - Personality becomes haughty.

This is noticeable in their dialogue, selecting a different spirit has a huge difference on how they interact with the bosses.

Wolf Reimu is straight-up nasty, bullying Eika/Youka to the point of tears:

Otter Reimu, on the other hand, is a little nicer about it, though perhaps uncharacteristically so for Reimu Plows-Through-Her-Problems Hakurei:

And then Eagle Reimu is extremely prideful, the possession inflating her self-image:

The actual mechanics of this spirit possession don’t seem to be very-well understood yet, but in stage 3, Kutaku has a lot of ominous things to say about the phenomenon. According to her, the Yama (known to be Eiki Shiki) is aware of what the protagonists are doing, and claims the spirits are untrustworthy, and that they’re known to exploit humans at the slightest opportunity:

All things considered, this doesn’t paint the spirit possession in a very good light, and things are seeming quite worrisome as the protagonists continue their journey into hell under the influence of these beings…