The Koopa kids are clearly Peach's. She and Bowser have a massive love affair, but for political reasons can't really let it be public. So they stage kidnappings every now and again. Its why there's absolutely no security for the Princess at all, even though she's been kidnapped dozens of times… and why they're only friendly enough terms with Bowser to go kart racing and playing various other sports.
Mario is not in on this, and thinks he's actually rescuing her. Luigi however, DOES know, which is why he frequently doesn't bother going to help.
Also, if you really want to play crazy continuity, check the old cartoon show... where the overall premise would be the same but the entire world (and Koopa) would change wildly between episodes, even so far as going into Star Wars territory... and for them to a couple years later introduce cavemen and Yoshi into the mix. And also that King Koopa, while obviously based off the oriignal Super Mario 1 sprite... looked very little like what Bowser came to be.