I would absolutely love the ability to see which pets i already have in the tooltip of pets (journal, cages or unlearned item) similar to how pettracker does it, with one minor change

Instead of showing the icon for breed i would love to choose between icon and breed text (P/P, H/S, S/S and so forth)

And i just realized Rematch needs some attention for that to work as well (i guess?)

I agree that some additional modes and/or customization will make sense at some point. I tend to collect max rare, without paying too much attention to breeds, so what I would tend towards would be the level of each pet, colored by rarity.