What does "Share Song" in Apple Music? I clicked Share Song, but didn't receive any feedback (what Apple Music did). 

Is there some kind of social network? Do I have a profile? Who sees the shared songs?
You should be getting a pop-up like the below screenshot:


Then you'd be able to share via any app you like. It will create a link to the song you've chosen to share.

Sources:
iOSHacker