Here is what I think.
Baby Bowser was Bowser. As a baby.
The Koopalings came from a wife, who is never shown in games or other media, but has served her purpose with Bowser, and Bowser abandoned her (sad, yes, but...).
All those little Bowsers you see in Mario Party games are minions with no relation to Bowser, who were turned into what they look like with Bowser's magic.
Bowser Jr. was also born from a wife, but not the same wife as the one who conceived the Koopalings. This makes Bowser Jr. the Koopalings' half-brother. Bowser then lied to Jr., telling him Peach was his mother.