in emergency/in case of emergency

They mean the same to me, as well.

Good to hear it. I just had a conversation with a friend of mine (a native speaker) and I used "in case of emergency" in a sentence. He said that it was rather unnatural and not used in American English. How come?

I interpreted your original phrases to be those used on an alert that one might carry in a wallet or see posted on the door of a building. Both are common.
Your friend is correct that it probably wouldn't be said in conversation. More common would be In an emergency or In (the) case of an emergency.

No matter how bad that is, I had no idea that "In case of emergency" sounded unnatural. Is it never used in everyday speech?

Most would not find fault if you used it in casual conversation. Don't worry about it.

I will avoid using it. I hope I won't forget though.