While I realize this is not the case, it is my understanding that, traditionally, what ensures a person is Jewish is being born from a Jewish woman, as (again traditionally), one could not prove who was the father. Of course, in the present age, that is no longer the case since we can trace DNA. That said, this particular case presents a unique situation since one woman donated an egg and the other donated her uterus for gestation. How much sicker than this can society become! This, IMHO, is such a violation of Jewish tradition and heritage. My empathy is for the rabbis who have to deal with this nonsense.
This ruling is from a secular court, so Rabbis don’t deal with it directly. You’ll note the condemnation of the process in the article, though irrelevant. The question of whether the child is Jewish would be an issue left to religious authorities. Presuming the “two mommies” are Jewish, I doubt it would be an issue. If not, quite a mess though I suspect the “birth” mother, I’ll call her the carrier, would prevail. But I admit I’ve nothing to base that on other than common sense.