Directed by: Aaron Horvath, Michael Jelenic
Starring: Chris Pratt, Anya Taylor-Joy, Charlie Day
Question: Why is the Princess called Peach instead of Toadstool and why is the main villain called Bowser instead of King Koopa?
Answer: Maybe the Princess's name was already Peach, but she just hasn't said since she was a baby. And Bowser doesn't just have Koopas in his kingdom; he also had other things, like all the yellow guys called by the name of Yoshi.
Deliberate mistake: Bowser has several teeth that appear and disappear throughout the movie. The best example of this is when Mario is trying to grab the Power Star. We see Bowser snarl at Mario before he picks him up. There is a gap between his two front teeth. However, when the scene cuts from Mario's horrified face to Bowser's face, there are now extra teeth inside that gap.
Princess Peach: There's a huge universe out there. With a lot of galaxies. They're all counting on us.
Answer: "Princess Toadstool" was used in the original English-language manual, but she was Peach in the original Japanese, and that name continued in later versions, being combined with Toadstool. Bowser was originally called "Kuppa", but presumably the English-language version resonated more with people, plus removed any confusion of "Kuppa, King of the Koopas", so the English version stuck.