Does Twining change your movement capabilities?

The important part is that the traversal kit is not yours, it belongs to the Aniimo. Each one carries its own mobility skills, so what you can climb, cross or reach depends on which Aniimo is active when you twine. Change the active Aniimo and you change your options for getting around.

Twining can be activated outside of combat, which makes it a field tool as much as a fight tool. You are not ordering a companion around from behind it either. The transformation puts you into the Aniimo directly, seeing the world from its perspective, and you drive its movement and use its mobility abilities yourself.

That reaches past movement into how you deal with the world generally. Ground that reads as a wall in one form is a route in another, so the question stops being where you can go and becomes which Aniimo you are going as.

The form you twine into is not always the same form

An Aniimo's appearance and form can shift with its region, its environment, the weather and the time of day. The body you are wearing when you twine may not look the way it did the last time you used it somewhere else.

Not settled: Whether those regional and weather-driven forms change what an Aniimo can actually do in the field, rather than only how it looks, is not settled.