Is it appropriate to call the circular edges of a cylinder "corners"? I recently corrected someone when they were describing a machining operation on a lathe. They referred to the circular edge of a cylinder as a "corner" and I got confused.

Then, thinking more about this, I realized that maybe it is indeed a corner mathematically speaking and maybe they have a better intuition about geometry than I do.
The ambiguity of the term 'corner' is why we tend not to use it in mathematics. Instead we talk about 'vertices' and 'edges'. A cylinder does not have any vertices, but it does have two edges.

Before deciding whether it is mathematically appropriate to call the edge a 'corner' you would have to define in terms of mathematics what you mean by a corner. Once you have done that, you could then start deciding whether or not the edge is a corner. Until then the question has no mathematical answer.
I don't think that's the correct terminology. For any figure of that sort we generally use the term vertex, edge etc. The question is more language based than mathematical. More essentially a cylinder has no vertices. So that may not be the correct term.