Transformations and change of basis in CG
You're correct, we don't usually worry about pulling the inner product through a change of basis in graphics. As a practical matter, calculations that depend on angles and distances, such as lighting and shading, are always done in world space or in a space orthonormally related to it (e.g. camera space).
The place where non-orthonormal transformations show up is usually in local-to-world transforms of meshes—most frequently with skeletal animation, where things get deformed, or where bones are set up with non-orthonormal coordinates for rigging/animation reasons. However in those instances we are always bringing the vertices, normals, etc into world space or camera space to process. We neither push back the world inner product into the local spaces, nor do we want to pull a locally defined inner product into world space, as neither of those is really useful for the typical kinds of things we need to do.
(And for the normals, we do know how to use the inverse transpose matrix to transform those properly, thank goodness!)
The other big non-orthonormal transformation in graphics is the projection matrix, but again, we don't really ever want to use the inner product in post-projective space, so the issue doesn't really arise. We just keep all our inner product-ing in world space.