In the original version of "Donkey Kong", Shigeru Miyamoto called the hero "Jumpman".
Nintendo of America renamed him "Mario" for America, since they thought that was more of a "New York construction worker" kind of name. Mario was also the name of the landlord of the person who suggested it.
Shiggy heard the name and liked it, so "Mario Bros" starred Mario (formerly known as Jumpman) and Luigi, who were both now plumbers. (I'm not sure if Mario was still known as Jumpman in Donkey Kong Junior.)
The Princess was called "Princess Peach" in Japan, and Bowser was just called "King Koopa" (the Koopas are those turtles). Both of these characters first appeared in "Super Mario Bros", of course.
NOA thought "King Koopa" was a little boring, so they renamed him "Bowser".
NOA was also worried that with Peach being a human and ruling over the "lower lifeform" fungus-based Mushroom People, they would get accusations of "apartheid" and "racial segregation" and crap like that (freaks and weirdos used to crawl out of the woodwork in America to try and attack Nintendo), so they renamed Peach to "Princess Toadstool" to suggest that she was also a mushroom. Not human, but one of the Mushroom People, albeit a very beautiful one (at least by human standards).
In Mario 64, the Princess invited Mario to her castle with a letter that she clearly signed "Peach". So rather than scribble that out and try to write something else, NOA met with Shiggy to ask what he wanted done, and they decided to merge the Japanese and American names.
Bowser is now "Bowser, King of the Koopas". With Bowser being his name, and King of the Koopas being his title.
The Princess is now "Princess Peach Toadstool". Peach is her first name, Toadstool is her last name.
BTW, Pauline (Mario's girlfriend from Donkey Kong), and Princess Daisy (the Princess of Sarasaraland in Super Mario Land on the GameBoy) are entirely different people.