How do I have OTHERS animal companions besides Frinos? And how do I send Frinos away?

[Sagaz]

I love the frog, but I wanted the kitten. Unfortunately the cat never appeared again and I don't know how to catch the white crow. Someone help me. I don't want Frinos with me anymore.

[Sentient_Toaster]

Toula the cat has moved to a specific location where it will always appear until you bond with it via one of the Witch's Delight familiar treats. (Location: At the pier you reach after defeating the first boss on the surface).

Raki the raven is trickier to find, because he only spawns in certain sort of room which isn't guaranteed to appear in any given run (Fountain room in Erebus).

[Juggen]

Make sure to have cookies with you since it's a random encounter, and I think you need to meet all animals twice before you can bond (although just one cookie).