TL;DR - Pullman has not said much about the basis for Dæmon forms. However from what we do know, Dæmon forms relate to personality traits and appear to be topical, they do not have a direct link to mythology. However, this is not to say categorically that no Dæmon form could be linked to mythology in some way - it simply means that there is no overarching consistency of source material from which the author drew upon in determining Dæmon forms.
The below is a link to a transcript of an interview with Philip Pullman that took place in Oxford, England, in August 2000. In this interview he was asked the following question: "what would your Dæmon be?"
In giving his answer (in which the author discusses variously how a human's personality is what determines a Dæmon's form), Pullman concludes with the following:
"But I do not know what my demon [sic] would be, and I cannot choose. The way to find out what your demon is, is to ask your friends to write it down anonymously. Then you will find out."
Pullman interview transcript, August 2000, Oxford England
This answer strongly implies that when creating his work, Pullman had no overarching or all-embracing logic in mind with regards to what form a Dæmon might take.
The suggestion that anyone would be able to attribute a suitable Dæmon form for someone they know, implies that a topical or associative basis for Dæmon forms would be acceptable. It suggests that "pop psychology" could very well be used in determining an appropriate Dæmon form.
The example of servants usually having dog Dæmons may well be indicative of this: dogs are a commonly-domesticated animal, frequently subservient to a master. A dog may therefore be seen as an appropriate Dæmon for someone whose personality is inclined towards being in a subservient position. (Or as Lyra explains to Will, such people enjoy the order of being part of a hierarchy, and of being able to please a person in a position of authority). Consequently we are led to understand that this is a case of people finding a place for themselves in Lyra's world that befits their personality.
It may be further noted that in the same interview, Pullman was asked about the sex of Dæmons, and what significance a Dæmon's sex held. (In most cases in His Dark Materials, a Dæmon's sex is the opposite of its human counterpart. However, there are some rare exceptions where same-sex Dæmons are observed.)
Pullman is asked about the rare instances of same-sex Dæmons and whether there is a link to homosexuality, to which he responds:
" I don’t know. There are plenty of things about my worlds I don’t know, and that’s one of them. It might do! But it might not! Occasionally, no doubt, people do have a demon of the same sex; that might indicate homosexuality, or it might indicate some other sort of gift or quality, such as second sight. I do not know. But I don’t have to know everything about what I write."
Again, this confirms the idea that there is no consistency of framework being used by the author; we can conclude based on these words that Dæmon sex, like Dæmon animal form, could be attributed on a topical basis.
There is a joking suggestion from the interviewer to Pullman that "you can make it up as you go along", to which Pullman replies that this is true
"To a certain extent, but then you discover the rules of the world that you're building as you build them... So there are other things, no doubt, I haven’t yet imagined or thought of about demons [sic]. I do find it’s a very rich idea, and right at the end of AS[Amber Spyglass], after 1200 or more pages, I was still discovering new things I could do with this human-demon link."
This explanation of the author's thought process further clarifies the position that there is no one overarching framework for Dæmons and their forms. Pullman drew on differing inspirations, and his ideas developed as the novels were written.