Relistening to Worlds Beyond Number's 'The Wizard, The Witch, and The Wild One' from the start (because I need it to live), and a striking minor detail stood out to me about the two occasions we have seen of a Witch getting her familiar.
I think it's been said explicitly either in an episode or as part of the patreon talkbacks that familiars are intended to be an embodiment of a fragment of the witch that she would like to internalise from herself.
But what I dont recall being commented on is that, not only do we see that on both occasions the Witch in question appeals to a spirit to aid in finding a familiar, but in both instances the resultant familiar shares some aspect of the spirit who provided said aid.
Daomai, who facilitates Ame finding The Fox, is very food focused when we first meet him; it could even be said that he's almost transactional about getting food before being willing to offer help. This is a behaviour that we see pretty prominently in The Fox for at least the first couple of chapters.
Eursulon goes on to SPOILERS aid Nif in finding the Elk (I can't remember whether he got a name). Eursulon is a spirit of freedom, but he's also very prominently a protector, defender and guardian throughout the show. It's my belief that the elk is supposed to be an externalisation of Nif's fear (she initially goes out searching for a prey animal more like a rabbit or a mouse) but filtered through the lens of Eursulon as a warrior, defender, and spirit of the forest.
Now, on a meta textual level, it could likely be that Brennan is trying to match the tone of the moment with how the familiars act as they first appear, which ends up influencing their ongoing personality.
But I really like this idea that the familiar granted to a Witch is in some way influenced by the spirit she appeals to for help in the process. It poses such interesting world building questions around which spirits granted the other Witches their familiars, why those spirits were chosen to help, and how they may have left their mark upon how the familiar turned out (not you Mirara, we all know that daddy-in-black brought you the angstiest loneliest goth bird).