Troops that come from factions have both their faction and associated kingdom on the troop card. Why don’t pets? Everytime I’m told I’m missing a pet to level a kingdom I have to do research to find that pet/faction I need.
Easiest way is to filter in the pet menu on the kingdom that you want to level. Then you can see the faction on the pet card front.
But indeed, it would be definitely better to have both kingdom and faction names mentioned on the card though, just like the troops.
Lifehack: just tap on the “Upgrade a [Kingdom] pet to level X” task in the Power Level requirements screen, and it will automatically bring you to your pet list, already filtered by pets belonging to that kindgom.
This also works for the other task types.