At the point the western half of the Roman Empire collapsed they were using a system with two emperors due to the massive amount of territory being impractical for one man to govern, senate or no. Only one of the imperial titles imploded, with the other going along just fine for centuries before that part of the empire also started to collapse.
Which claimants are you thinking of? I know the Ottoman Empire and the Russian Empire both claimed to be continuations of the Roman Empire. I don’t think Italy ever claimed to be the new Rome, somewhat ironically, and I think Germany and France had stopped claiming to be Rome as well.