So are there any studies of people who kidnap children to raise and how, if any way, their relationship to the child, is different from that of birth mothers?

Like, the Judgment of Solomon involved a false mother who had had a baby of equal age that she smothered in her sleep and then claimed the baby at issue as a replacement, that might change things (like, how she weighs the identity of baby vs. having something) but would a reverse-cuckoo mother otherwise be any possible to “detect”?