If the superpowers-as-puberty message wasn't intentional in the recent movies, it sure was back when they were being developed. For most of Spider-Man's 49-year history, Peter Parker lacked the ability to shoot webs out of his own body (heh), instead relying on a pair of mechanical shooters that he built himself and attached to his wrists.
This new power of Spider-Man's is prominently featured in an unused script written by James Cameron in 1991. If you still have any doubts about the subtext that's going on here, check out this scene from Cameron's version, in which a newly spiderized Peter Parker wakes up in bed:
Something is causing the sheet to stick to him. He lifts it, revealing a sticky, white mass completely covering him, gluing him to his bedding. It is some silky substance webbing him into the covers. He cries out in dismay ... struggling to free himself from the gluey strands. Where did it come from? He notices his wrists. ... They are oozing a pearlescent white fluid from almost invisible slits about a quarter of an inch long.
See, it could have been much, much worse. It's pretty clear that this metaphor was a deliberate one. Either that, or Cameron has a few things that he needs to work through with his therapist.