I think it comes from Date Of Birth - DOB. I think legally, the police/courts don't have a contract until they get your date of birth. They can do this via your name. But if you won't tell them, not a lot they can do 'cept throw you in a cell and hope you talk eventually. However, if your friends, neighbours, or someone else who knows you rats you out, that's called dobbing.Dobbing! What an awesome word.
I’m assuming that basically means “ratting out” or tattling on your neighbors?