Why do human babies regularly spit up? What purpose does this function serve?
I’ve never seen another infant mammal do this on a regular basis - do others do it?
I believe this has to do with human babies being very, very “unfinished” or premature, if you wish, as compared to other mammal babies.
Ya’, and it took my son 14 months to become “finished.”
Why do you assume it serves a purpose?
Why would every baby do it if there wasn’t a reason for it? 
You’re laboring under the “evolution has a goal” fallacy. Genetic characteristics are only selected out if they result in your death before you can reproduce. Apparently, the spitting up function has not significantly affected human babies’ ability to survive to reproductive maturity, so it was never selected out.
IANAD, but I can think of various reasons why spitting up is a necessary response. If the baby cannot digest everything it has eaten, spitting it up would be the only way to get rid of it. Producing too much digestive fluid may cause it also. Irritation of the stomach lining or the valve opening into the stomach can cause it. It’s mostly an involuntary response to a variety of situations.
If a baby spits up soured milk, it’s probably been in the stomach and partially digested. If a baby spits up unsoured milk, it may not be making it into the stomach at all (which may indicate a serious condition).
14 months it usually takes at least 20 years! 
Hard to argue that point!
Not every baby does. Breastfed babies spit up less than bottlefed, presumeably because there are no air bubbles in breasts. Some kids are less “burpy” than others as well.
Our son is 1 and has spit up maybe five times.
Now, his farting on the other hand…
Cyn, Postpartum RN
Every work day I handle newborns aged 2 hours to 3 days old. Even breastfed babies spit up. Not all babies spit up. Some babies are gaggy and spitty due to swallowed amniotic fluids. Some are air gulpers and some have milk allergies. Not every baby has an effective suck, be it breast or bottle fed.
I remember seeing kittens spit up. I was staying with my sister when her cat had a litter and the would spit up occasionally.
Of course this was probably just practice for hairballs. 
Jim