You remember wrong. It just means "of rome" and it's that because it was a roman province. There's a difference between "of rome" which is what "român" means and "the true descendants of Rome" which is what you're claiming. We've never claimed to be more Roman than Italians or anything. That'd be stupid.
According to the anthem, romanians claim to have the blood of the romans and the name of Trajan, meaning that everyone in Romania had fathers that had fathers that had fathers... that were the sons of the roman emperor that conquered Dacia