Timothy Chow gave a fine answer in the context of classical mathematics. Here are some further sources for you to ponder. These not only work without choice, but also without excluded middle:
Homotopy Type Theory: Univalent Foundations of Mathematics. For instance, is Blakers-Massey theorem advanced enough to count as real math?
C-CoRN library, skim the README or see this paper for a humane summary of what is in it.
UniMath library, inititiated by the late Vladimir Voevodsky, browse this folder to get a feel for what is in it.
The moral of the story is that the folk tales that mathematicians tell about how choice-free mathematics is completely different, or even impoverished beyond recognition, are just folk tales. Of course, one has to be a bit more careful, but that is always the case when we generalize, as new phenomena arise.
