In QFT or CFT, say the action is invariant under some local transformation. Can we call that transformation a Gauge transform?
There is a specific notion of gauge transform in math which is defined as $G$-equivariant diffeomorphism from some principal bundle to itself with some specific properties.
Are physicist gauge theory and mathematicians' gauge theory the same thing?
No, what physicists and mathematicians mean by gauge theory are not "the same thing", but of course there is a reason the mathematical subfield is named after the physical subfield:
You can phrase many physical gauge theories in terms of the language of principal bundles and mathematical gauge theory more generally. For instance, you can read me doing this here for generic Yang-Mills theories, here specifically for the notion of large gauge transformations, here for general relativity and here for Donaldson invariants.
Most "normal" physical texts will not talk about gauge theories in this manner, choosing to work only locally and always in a single coordinate patch. There are no bundles there, and none of the global constructions and invariants mathematicans care about play any role at all at the introductory level. An understanding of mathematical gauge theory does not automatically empower anyone to understand physics texts about gauge theories and vice versa.
Additionally, while the Lagrangian formalism at least in principle directly corresponds to the mathematical idea of gauge theory, the Hamiltonian formalism of gauge theories is much unlike that even though it should be physically equivalent. Here the kind of mathematics that matters is instead symplectic geometry and in particular symplectic reductions, not "gauge theory", see e.g. this answer of mine on the relation between Lagrangian and Hamiltonian gauge theories and this answer of mine on the physical relevance of symplectic reduction.
