Why did this develop/persist in Ukrainian and not other languages?
Your premise is not true.
To start with, English also has special verbs describing giving birth to different animals:
- farrowing (of a sow giving birth to piglets)
- foaling (of a horse giving birth to foals)
- calving (of a cow giving birth to calves)
- lambing (of a sheep giving birth to lambs)
In Ukrainian it is equally easy (if not easier, but I am biased!) to form verbs from nouns. ощетинитись means obtain a bristle [щетина], which in English means to bristle (also a noun acting as a verb). Other forms: заколоситись from noun колос, or оголосити from noun голос (to voice).
So, having different verbs for giving birth to different animals is both common with other languages, and natural.
- опороситись (from порося = piglet)
- ожеребитись (from жереб'я = foal)
- отелитись (from теля = calf)
- ягнитись (from ягня = lamb)
- окотитись (from кошеня = kitten) - although this one is confusingly used for lambs and goats as well (for example, in a traditional Christmas song вийди, вийди, господарю, подивися на кошару: там овечки покотились, а ягнята народились)
- ощенитись (from щеня = puppy)
Those verbs are formed from the names of baby animals, for example:
- Опороситися 'to piglet oneself'
- Отелитися 'to calf oneself'
- Ягнитися 'to lamb oneself'
- Щенитися 'to puppy oneself'
Agriculture has always played an important role in the life of Ukrainians, that is why such verbs are important. Still, all of them can be substituted with a generic verb народити 'to give birth'.