OK, so only mammals – and even then only placental mammals even have umbilical cords. Those that do all need to “cut” them, other animals use their teeth.
Humans just have become averse to using their teeth to do it… and we also frown on eating the afterbirth, but that is common, too, for placental mammals.
What about the other animals? Well, marsupials just don’t have placentas and thus are not connected to them with umbilical cords! They are birthed very, very underdeveloped and get nutrients instead from a teat in a pouch until they are ready to be out and about.
Birds and reptiles don’t have umbilical cords, either, but instead absorb the yolk of their egg. In fact, one of the last stages of development is to “zip up” around that last bit of yolk and in a newly hatched animal like this turtle you can sometimes still see the last of it poking out:
But only mammals that have a placenta have an umbilical cord – and if they have it, they need to cut it, and they use their teeth to do so. Humans just decided that we would rather use a tool than our teeth because humans are weird like that.
