Okay I know he's big, scaly, has wings, flies and breathes fire, but no, really he's not a dragon. Don't forget that Pokemon is an Eastern invention, and Charizard only looks like the Western version of a dragon (I really do need to do a post on the difference between Eastern and Western dragons.)
There's also lots of Pokemony reasons why he's not a dragon. Look at this amazing video by "The Pokemon Master" who can explain things in better detail than me.
Of course I'm still going to pretend he's a dragon when I'm playing as him in Super Smash Bros :)