If he was a dragon, he would have leathery wings, but he has feathered wings, hence he is a dinosaur (since only dinosaurs have feathers). He also lays eggs shaped like bird eggs, which also indicates he is a dinosaur. Plus his two-legged stance with legs directly under his body and smaller forelimbs useful only for grasping (at most) is characteristic of therapod dinosaurs, while most dragons you see walk on all fours (although they might be able to use their front limbs for grasping on occasion, there front and rear legs are much closer in size).
Also, if Yoshi lays eggs, doesn't that mean he's actually a she? All yoshis seem to lay eggs, so they must either be asexual (which means yoshi is a she) or hermaphrodites (which means yoshi is really both). We can't tell which because we never actually see Yoshi lay an egg with another yoshi in it as far as I am aware.