Actually, now that I think about it, every SHIELD character has no backstory, and what little has been eluded to is purposely kept foggy in order to add mystery to the character. Even Coulson is a complete mystery, as to who he was before he showed up in the Iron Man films, and he has his own TV show. The extent of his backstory there starts with the films and occasionally bumping into people he knows from past SHIELD operations.
I think an example to look at is Hulk. The Phase 1 set of movies was mostly unrelated to any previous films, but there had been a non-Marvel Hulk film recently enough that people didn't need an origin story in the Edward Norton version. Then Mark Ruffalo Hulk shows up in Avengers and there is roughly a 30 second conversation on where Banner has been all this time. The only confusion they caused was some people couldn't understand how he can be completely out of control at one point and then fully in control at another point.
If Peter Parker just shows up everyone will know who, what, and why. Add in a quick expository conversation about the spider bite and move on.
If it is Morales people will be confused and want to know how he wound up with the spider powers.