Understanding these songs, I guess, depends on where one was raised. We used to sing this song, and other western songs in school, and I can't remember ever not knowing the meaning of the words. School mates included some from the farms,ranches and sheep ranges.

Of course, Old Dan is ambiguous; I tend to think of him as a pack animal, as mentioned above.
He could be a horse, donkey or mule, but I doubt that a burro, Equus asinus (under BLM management, there is an adoption program), a small creature originally from Africa and brought by the Spaniards, could have been of much use. They were ideal for small farmers, prospectors, vendors in town selling piñon wood, etc. because of their hardiness.
I bought one for 50 cents when I was in grade school- my parents were most upset when I brought it home.

Some people call a small donkey a burro, but as a New Mexican, I object!