Peter has had a run in with all of your people. I know there's been a couple of crossovers where X-23 and Spidey have run into each other. I remember one pretty early in her run that was fun. So he'll definitely remember that at least.
That said, I never did read any of the Infinity Warps stuff, yet, still he'll at the very least recognize her!
No explanation indeed. Peter loves himself some Storm, he's also a bit terrified and intimidated by her.