I think the story reboots every few games, so it’s not like say, the Mega Man games where every game is part of one big continuity. There’s a setting and recurring characters that’s built up over the years and that’s about it; everything else is specific to that game or subseries. Basically, the Bombermen (M/F), who may or may not be siblings, are some kind of space police from the planet Bomber and they have to fight a villain, usually but not always Bagura/Buggler, to protect the peace in the galaxy.
There is a bit of a rabbit hole (puddle, really) you can go into where some of the earlier games have a connection to the Lode Runner games, because Hudson Soft did the Famicom port of Lode Runner. What it boils down to is that Lode Runner used to be Bomber Man. This connection hasn’t really been relevant for a long time, but the fact that Lode Runner is a Galactic Commando may have influenced the current setting.
It’s kind of hard to tell because it’s a low-poly model at a low resolution, but it looks like they’re wearing the mask kuroko wear, who are stagehands in Japanese theater that you’re expected to pretend aren’t there while they move things around on the stage.
This is from Tomodachi Life, right? I haven’t actually played it, but from what I know of the game, all the characters are Miis. So if the store staff were also Miis, there would be nothing to distinguish them from regular characters, so maybe they’re trying to avoid giving the impression that you can interact with them in the same way as you can with regular characters.