I’m told nipples are “made of” (if that’s the right term) erectile tissue, same as penises ( I always want to say “peni”, but I digress). Difference is, nipples do not have a vascular setup which causes them to become engorged with blood when aroused.
What we’re talking about is a hardening of the flesh in response to stimuli. To go back to the example of the penis, have you ever observed the changes in the flesh of testicles to weather, arousal, etc.? I think, again, that if not for the vascular setup in the penis that causes it to become engorged with blood so that an erection can be maintained through coitus, that the penis would also shrivel when aroused.
As to the “pop up” comment, I’d like to add that my nipples do not become larger when stimulated; they in fact shrink as the flesh tightens up. The reason people say that nipples “pop up” is that they now cause a “bump” in one’s clothing where there might have been a smooth line beforehand.