In Bart's previous game, he knew a version of MCU Peter Parker pretty well for a while, but also met another version he didn't really know. While Bart mostly doesn't care who knows his identity, he actually really respects others who keep the secret. He and Previous Parker actually half-jokingly made up a codename of Velcro-Lad because everyone in that game could generally recognize PCs and would ask what their powers were, and the only one that Pete wanted to admit was being sticky since it seems pretty lame on the surface.

Of course, if you want to avoid that kind of third-wall stuff, I'm happy to retcon that because...yeah.