Yes, it's "context sensitive" as you use the term.
And you can safely assume it should work with any multitexturing capable openGL driver/videocard (obviously that is not broken, of from some random guy from the garage)

"Because its OpenGL.
When something works for you it doesn't mean that it will work for everybody."

This can be truth, but not with a well defined thing like this. I mean the multitexturing states' "context sensitiveness" (as you use the term) is a so simple and default behavior, vendors won't fuck it up.

(I hope I'm right, someone correct me if my statements aren't true)