I found several increment errors while testing:

Skullwalltula and Big Skulltula is incrementing Skull Kid instead and skull kid increments skullwalltula.
Blue bubbles are incrementing Green Bubble instead and vice versa. White bubble increments red bubble
Gold Skulltulas are incrementing Big Skulltula instead
beamos increment biri instead, biri increments big octo, big octo increments beamos
flying pots increment flying tiles and vice versa
stingers do not increment enemies killed at all in the boomerang chest room
dark link is not incrementing anything

Are these errors present on develop too?

It seems to be bugged in completly different ways:
skullwaltulas track as gold skulls, gold skulls track as big skulls, and big skulls still track as skull kid, for example.
Seems like a mess and someone's going to have to go through and double check everything.

I'm testing on develop and I haven't gone through everything yet, but so far all the issues you have mentioned are showing up there, too.
I think I should fix the issues, then treat this as a bug fix and move it to develop-copper.

I'm gonna guess that most of these are simply misalignments between counters and labels. I can already confirm that to be true for the following:

Blue bubbles are incrementing Green Bubble instead and vice versa. White bubble increments red bubble

beamos increment biri instead, biri increments big octo, big octo increments beamos

Are the enemy names meant to be kept in alphabetical order? I'm hesitant to change the order of the enum values because that'll mess up everyone's enemy defeat counts as opposed to just switching the labels.
I'm also wondering why we aren't using a table macro for the count/name mappings like we do with everything else? What we have instead is an array for the names, intended to be indexed by the enum but the names don't all match up.

@JordanLongstaff cleaning up the code to be based less on implicit indexing would be good, but LGTM