And often physicists (as you and Legion have been noting) simplify to the point of lacking mathematical distinctions, which leads to a lack of nomenclature that mathematicians would use.
For instance this leads to just assuming a mathematician using a word in an unfamiliar way is a new concept (“oh, one of those math people things”) rather than the same thing we physics people know and love under a different or extended name.
And it is true that mathematicians have different goals. We tend to explore the abstract ideas and not worry about applications. We also require a much higher standard of proof of the ideas.
Physicists, on the other hand, are more interested in the real world (as they should be) and the strict mathematical idea of proof is unnecessary simply because the ideas have a different standard: that of testability and agreeing with observations.
But what I have found is that physicists tend to use the ideas and notation that were common in the late 19th century when doing general relativity. That was overly concerned with components of vectors and tensors. In the intervening century, mathematicians have cleaned up these ideas, simplified them, and unified them in ways that, I believe, could greatly help the discussions in physics.
One example is that of the Christoffel symbols. For most mathematicians, these have been replaced by the notion of a connection. A pseudo-Riemanning metric will select a particular connection for which the metric is invariant. But, and this is relevant in QM, there are other connections than this one and those can be relevant for modeling particle fields.
And, again, it isn't how the components change under a change of basis that is the key: it is a coordinate-free description that simplifies the analysis and clarifies the ideas. The real key is the differential operator that arises. Then, after you choose a coordinate system, the components can be dealt with easily.
Think of the simplification obtained by doing Maxwell's equations in vector form as opposed to writing out the formulas for the different components. Instead of four nice equations, the component version has 8 equations that are not nearly as compact nor clear. It gets even better when spacetime tensors are used: Maxwell's equations then reduce to two tensor equations.
And, once you have that, you can choose the coordinates you want, similar to deciding to have the E&M wave be along the z-axis. it simplifies the analysis and loses no generality. But, you don't worry about how the components would change if the direction was along some other vector. You go back to the vector equations and write them down directly in the new coordinate system.