"What kind of a song?
exactly! There are as many ways to define 'categories' as there are ways to define words. They can be totally subjective, depending on how YOU think of them. You could divide everything into 'fast songs' and 'slow songs', or as some fiddlers do, into the key-- "I know some good 'A' tunes. Or into topics...(our monthly sing has a topic)...or by length. etc...etc...
But the choosing of categories changes if you are planning a public event or making shelf signs in a store, as there are general categories that folks look for when going out to buy music.

Everything can be divided in similar ways, depending on culture, specific needs, ...and the very language involved. What is important is to agree on words and categories that allow some sort of reasonably clear communication. (In some places, you need to clarify what you mean when you invite someone for 'dinner' or 'supper'.)

The enemy of clarity is over-generalization and weak vocabulary....wars have been fought and religions altered because of confusion about what was intended.