> How much of the identity of a mathematician is tied up in being the first to discovery?

Here is how I'd put it: math has an enormous focus on discovery. It is why we have Godel's incompleteness theorems, the Cantor set, Zorn's lemma, and so forth. We name things after their discoverers.

It is possible that, going forward, no more things will be named after human discoverers. The last such naming (of something truly significant) may already have occurred.

That is a massive culture change, at the very least.

This has already been shifting in science and engineering because typically large teams are needed to get things done. Math has held to it better, but even there they are almost always multi-named concepts, like Rivest-Shamir-Adleman etc. And even those often have to ignore names that could plausibly be added.

For example in classic image processing / computer vision we had things like the Hough transform, Canny edge detector, Sobel filter, Shi-Tomasi detector, Lucas-Kanade optical flow, Horn-Schunck optical flow, Kanade-Lucas-Tomasi tracker, Otsu thresholding, Viola-Jones detector, etc, but nowadays we no longer name things like that, but typically with some abbreviation or some name that relates to the concept more closely.

It's in part the larger teams thing, and also a cultural trend to de-emphasize "great men theories".