Do all unmarked police cars have external traits that help identify them?

They do exist. There is some difference between the vehicles that "plain clothes" officers drive and those that "undercover" officers drive. I work in an undercover narcotics unit, and the vehicles they drive are 100% unmarked. No radios, lights, nothing. They are mostly just citizens' cars that were seized and then used once the case was adjudicated. If the officer is serving in a true undercover capacity, then you will not see them driving a car that is identifiable in any way.