I agree with Mewtwo's placement, Bowser, not so much.
He's got losing (and in some cases, unwinnable) matchups pretty much across the board. Bowser loses to Jiggs (due to the ludicrous ease of resting him), Olimar (for the same reason that Bowser loses to other characters with long grabs and huge disjoint), ICs (their combos on him are absolutely insane, plus his command grab and dash grab are totally nullified, and their projectile appears to have been designed specifically to screw bowser), and to Ganondorf (whose aerials are totally safe on Bowser's shield, and has a command grab that stuffs most of Bowser's options while leading to insane kill combos).
CF, Ivysaur, ZSS, Link, and Mewtwo are effectively guaranteed losses for Bowser.
In short, Bowser loses to all the other consensus "terrible" or "bottom 5" characters, and it hardly gets better as you move up the tier list. He
does have a couple even or positive MUs (such as Falco), but you could probably count them on one hand.
Someone has to finish last. What makes you so incredulous that it might be bowser? Who could be worse?