they just don't need to name roles like the average 9-to-5 worker;

Tony never said those exact lines - "I'm bumping you and Sil up to caporegime and Consigliere respectively" - but I believe that's what he meant.

they just don't need to use the word 'underboss' to fill job reports but what I assumed is that by the beginning of Season 2, when Tony undoubtly takes over as "street boss", he chooses Paulie and Silvio as his #2 men, no matter what you call it. Silvio is brighter and more level headed so he's the advisor and Paulie is the only other one Tony can trust so he's the one doing what an underboss does.