That's an interesting question! It's hard to say but I think the daemons are the sort of "essence of humanity" (confusingly, as they present as animals). Even the intelligent talking animals in Lyra's world don't have daemons, e.g. the giant bears. The witches however do, which makes sense as they are part-human.
We also know that the daemons are tied to to Dust/Sin - so the children who have had them severed are able to live, but they are reduced to shadows of themselves - little more than automatons. They have lost their essential "humanness", their ability to feel true pleasure or really any emotion other than loss.
We also know that daemons are sort of linked to, for want of a better word, sexuality and sexual maturity - they take on their fixed forms once they've been touched by a lover (I feel a bit weird typing that but I think it's how it's explained in the Amber Spyglass).
So I think that overall they represent the quality that humanity has that sets us apart from animals - our intelligence, our ability to feel very strong emotions, to love, to create art, to create meaning in life
Not sure if that all makes sense, I'd be interested to know what other people think they represent.