"Home" in English can mean both where you're currently living and where you came from. So when you're at a university, sometimes they ask for your current address and your permanent address. "Home home" would be your permanent address since where you have more stability at. So as Peachey said, it just adds emphasis that your "home home" is where your heart truly is, but your "home" is where you currently live.

It's something that people do with casual speech. If a word means a wide range of things, it may be repeated to indicate that the speaker means the more permanent or deeper meaning. So yes, you hear things like - "My guitar is at my *home* home." or "I know you like him, but do you *like* like him?" or "It's mostly done.... but it isn't quite *done* done yet."

"Home home" means your real home or actual home. This is not standard English, but we use this repetition in colloquial speech.