It is. Most people don't have a 20% project though. It means less time to work on the things that might get you promoted.
But you can have one of you want one.
But you can have one of you want one.
A common reason people do 20% work is to test the waters of another area of the company. It's also a kind of tryout for you.
You do tell your manager in this case, yes. They're not supposed to say no. Hard to say what the range of responses is in practice. In my experience managers are supportive of this.