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In this series the first set of omnibuses is about is about a female protagonist who has the bloodline of two of the kingdoms. One of the kingdoms has animal companions that are linked by birth to them when they're born, and only the linked person can communicate with them mentally. However, the protagonist, due to their bloodlines, is able to talk with any companion mentally. Eventually she marries someone from the final bloodline and her child is the perfect being and has a lion as their spirit animal.

I don't remember whether the protagonist has an animal companion or not. I believe at one point she develops amnesia (which is how she meets the people of the 3rd bloodline, and is whom she marries in the future), prior to this she had a relationship with someone from the other 2 bloodlines. I may be wrong about it being just one female protagonist up to this point, the female protagonist might have married someone from the first two bloodlines and have had a daughter/son who marries into the third and gives birth to the super baby.

I believe one kingdom lived in the forest (the one with spirit animals/animal companions), and the others on either side of the forest, maybe having a border the met outside the forest as well.

I read this about 10-13 years ago, and is not the Astor Chronicles. The book themselves seemed relatively new at the time ( < 2 years old), however since it was an omnibus of books, that means the original series had likely been out for around > 5 years at that point.

I wonder if this is the Cheysuli series by Jennifer Roberson.

If so the Cheysuli aren't exactly linked to animals, they can change into an animal. As I recall their animal shape is determined at birth so I guess this could be interpreted as being linked to a spirit animal at birth.

If I'm correct the protagonist is Alix and her adventures are described in the first book Shapechangers. As you say she has two bloodlines because her mother was a princess of the Homana people and eloped with one of the Cheysuli. So Alix is half Homana and half Cheysuli. Alix has the power to become any animal, which sort of tallies with your description.

This doesn't really fit with your memory of mentally communicating with the animal companions, but there are lots of other similarities. There are lots of books in the series and they have been collected into several omnibuses. There is a third bloodline, the Ihlini who are sorcerers. And there is (several books along) a super baby, whose adventures are described in Daughter of the Lion.