In the pet archive there's a new button "show completed months", which'll show you which months and years you've got a complete collection for. You can also filter the results by species or artist.
It shows a blue square next to months where you own some of the pets, a red cross where you don't own any of those pets, and a green tick where you own all of the pets from that month. It'll look like this:
The 'store pets' categories don't get counted in this total because we consider these to be separate from the regular pets.
You can check this feature out now in the Pet Archive: archive/2021/January/
EDIT: I've now added support in the Item Archive too!