I can't tell what kind of tree it is either, however, it is the shape that a lot of fruit trees have, so I think that you could use a fruit tree in the garden along with the flowers and veggies that you grow there.
It seems to me that if a tree is just going to produce shade, or it is going to also produce a crop of some kind (fruit, nuts), then it is better to have a tree that also gives back something useful besides shade. Plus, they usually have more blossoms if they produce fruit, although some are just flowering varieties, and those are pretty, even when they don't give you any fruit.
I think if you just find a tree that you like that will have this general shape, and is hardy where you live, it would probably work fine for the garden.
