Hundreds of public figures get Nobel Peace Prize nominations every year. Any university professor or politician serving at a national level can make a nomination. There are no requirements for who can or can't be nominated; there is no "official" selection process in which some nominees are accepted and some aren't. All it takes is precisely one university professor or national-level politician that's willing to make the nomination. It's not at all extraordinary that the President of the United States, the most famous, powerful, and influential person in the world, has met this very low bar. I can't prove this, because the Nobel Committee doesn't release lists of all nominees, but I have no doubt that every president since the formation of the Nobel Prize has received at least one nomination. The president is simply too big to be ignored.
I understand why an egotistical idiot like Trump is celebrating this news as if he's already won or achieved...well, anything, and also why his staff are all presumably too afraid to try and set him straight and inevitably suffer his wrath for not telling him what he wants to hear. But for those of us living in the real world, this means nothing. Can't wait to see the angry tweetstorm from Trump when he doesn't win it, though.