Why can't a pterodactyl be a bird instead of a dinosaur?

Pterosaurs (what most people classify as pterodactlys, even though pterodactyls were only one species) were neither birds nor dinosaurs. Birds evolved from dinosaurs, and pterosaurs evolved from the archosarus, an ancestor common to both pterosaurs and dinosaurs. Pterosaurs are thought to have been covered with hair. Their wing structure is different from that of a bird. A bird's flight surface is made up of feathers, and a pterosaurs is made of skin streched between the femur and an elongated fourth finger.

Simply put, pterosaurs can't have been birds because they are in no way related to birds, and they weren't dinosaurs, though they and dinosaurs evolved from a common ancestor.

Answer2: Reptiles That Fly

One intriguing type of ancient reptile was the pterosaur ("winged lizard"), which includes the pterodactyl ("winged finger"). But these were not dinosaurs, nor were they birds. They were flying reptiles and are classified with other reptiles such as dinosaurs and crocodiles. Some of them had wingspans of 25 feet [8 m]. One discovered in Texas in 1975 indicates that some had wingspans of more than 50 feet [15 m]. These were perhaps the largest animals ever to fly.

While pterosaurs had the teeth, skull, pelvis, and hind feet of a reptile, they in no way resembled the reptilian dinosaurs. And while they appeared to be a bird with stiff aerodynamic wings, they were much different. Like birds, the pterosaurs had hollow bones and few flexible joints in wings and ankles. However, a bird's wings use feathers rather than a membrane as was the case with the pterosaurs. And the fourth finger of the pterosaur forelimb extended to support the wing membrane. In the bird the second finger is the principal support of the wing. Excerpt from Awake magazine Jehoovah's Witnesses official website.