> 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.