This sentence is so... cozy.
Should "tell me a story" also work, since in both cases its an implied "you" subject?
No, because your sentence is a command, not a description, like the given sentence.
I mean "you, tell me a story" would be a command... If it's purely meant as a statement then shouldn't it be "you are telling me a story"?
Depends on the way it's said, surely? Tell ME a story vs Tell me a story? I can't see any reason that 'Tell me a story' is any more or less of a command than 'You tell me a story'. Both depend on how they are said.
Would narrate be also correct here?