I'm not a lawyer, but: it becomes a crime when you use it to commit fraud. If you fake a name to buy something embarrassing, then pay the other person as agreed, nothing illegal happened.

My understanding is that using an alias to commit a fraud is an aggravating factor, like it means you went out of your way to deliberately hide your crime. It'd be awfully hard to convince a court that you made an honest mistake if you lied about your identity during the process.