I own the TurboGrafx 16 version, it was one of my favorite games for it. That was my only console as a child, good times. There is a cutscene at the start, and that’s all the lore the manual offers too: https://r.mprd.se/TurboGrafx-16/TG16-PCE/EXTRAS/Manuals/Bomberman%20%28U%29.pdf

Something along the lines of a inventor’s daughter being kidnapped by an invention gone evil (black Bomberman), and you are the one who didn’t go bad, trying to rescue her (which you do by the end).

I don’t have the manual anymore, but it’s findable online.