Unity only licenses their game engine to us, which makes sense.
This site (https://www.legalzoom.com/intellectual-property-rights/copyrights/protecting-and-selling-your-mobile) says that I’m not allow to copyright the source code of my games if it uses any open source code.
I know Unity is closed source but do they have some kind of similar licensing terms? Like, I’m only allow to copyright my assets, game music, story, etc. but not the code?
Copyrighting source code is not the same thing as copyrighting your game. The actual code itself is only one part of your game, which includes art, possibly writing and music, and has an automatic copyright on it that applies to the game as a whole at the moment of creation. So any game you make in Unity is copyright to you. That would allow you to attempt to sue someone who stole your art, or music, or plot. Suing someone because they stole some of your source code would be a different matter.
This article is long, but it explains the matter in great detail: What an Indie Needs to Know About Copyright – GBGames
You can copyright your code in Unity so long as you don’t use any other plugins that are licensed in such a way as to forbid you from doing so. If you buy or download code assets from the Asset Store then they will not have a license that restricts you.