There is one point I have to address. I can't resist clearing a few things up.
They're not even in the same class! One of them is a reptile, the other's bird! Let alone the same genus!
It is true, using the Linnaeus system of taxonomy that was created over 200 years ago, Birds are in the class Aves and Dinosaurs are in the class reptillia. Dinosauria is a super order. So, how can Birds be in a super-order that belongs to a different class?
Because of this:
Linnaeus system of taxonomy that was created over 200 years ago
The Linnaeus system was created before researchers actually uncovered all the data that shows the evolutionary history of birds. Linnaeus just straight up divided all animals into six classes, based on current observations. As a system for classifying current species, it works all right. But if you want to compare the relationship between two species, it is horribly outdated. Birds are more closely related to any reptile than any non-reptile. Therefore, birds should be in the same taxonomic group as reptiles, since they split off at a later date than say, mammals. And all vertebrates are more closely related to fish than non-vertebrates, which his system also fails to demonstrate.
Any classification system worth anything has to be more than arbitrary words, like "avian dinosaur". Nowadays, words like dinosauria refer to things that are more related to each other than to non-dinosaurs. For instance, any dinosaur is more related to a modern eagle than a pterodactyl.
That is why we call eagles dinosaurs. Dinosauria marks a branching off on the evolutionary tree a long time ago. Think of current classifications as nodes on the tree, not as segments of a branch. Everything after this node is a dinosaur. This is known as monophyly.
In short: strict rank-based Linnaeus taxonomy is horribly outdated. A more modern method is phylogenetic nomenclature, which serves to not only classify organisms in an organized manner, but also reveals their evolution history and degrees of relation to other species.
The Linnaeus system is still taught largely because it is convenient until, well, it is no longer convenient. Like now. For instance, if I wanted to talk about different trees in my yard, I could use the Linnaeus system, because people don't care about the cladistics of these trees. But if I wanted to determine which were more closely related to each other, I would probably abandon this system and switch to cladistics.