kinda!! but,, kinda not,, I spent Some Time digging and didn't find much, but,, here's what I do know
so during Roman triumphs (a quasi religious festival where a general who was uniquely victorious would get to break a bunch of taboos during a festival to celebrate those victories) one custom is that the soldiers (who were last marching in the parade and who had spent years if not decades fighting with this usually their first time returning home / visiting the city) would entertain the crowd with crude songs poking fun at the general. we only have record of one, which was sung during Caeser's triumph
"Urbani, servate uxores, moechum calvum adducimus.
Aurum in gallia effutuisti, hic sumpsisti mutuum."
"Oh Romans watch your wives / we've brought home a bald adulterer (Caeser) / who pissed away all your gold in Gaul / and now he's come to ask for more!"
but that's not really a drinking song... even if it is rowdy, unfortunately history typically only records what people of the time value enough to maintain, and music was seen by Romans with the same derision that like,,, streamers are nowadays, so we don't really have many examples. Graffitti though! bar graffitti is every day folk saying fuck it I want this to be recorded and perserved and random chance lets them even if historians would have rather it not
a cool bar graffitti poem example
Nihil durare potest tempore perpetuo;
Cum bene Sol nituit, redditur Oceano,
Decrescit Phoebe, quae modo plena fuit,
Ventorum feritas saepe fit aura levis.
Nothing can last for ever;
Once the sun has shone, it returns beneath the sea.
The moon, once full, eventually wanes;
The violence of the winds, turns to a light breeze.
worth noting that Roman songs did not rhythm, rhythming did not make its way to Europe until the 6th century, melody was instead sung against itself in fifths (a modern example would be the tradition of tvísöngur in Icelandic music)
also worth noting that most of the Epics we think of (Aeneid, Odyssey, ect) were sung, and popular lines from those poems were found as graffiti, so it's imaginable that really kick ass or famous lines of it were likely used as drinking songs. I like to think of a bunch of folk drunkily shouting/singing their favorite parts of the story. my favorite example of people quoting these epics is like,, so the first line of Aeneid is "I sing of men and arms" and one graffiti was found with the line "“I sing not of arms nor man, but of fullers and owls" (fullers = laundry workers, owls = the symbol of Minerva, the god that protected fullers)
this! is such an interesting question! im gonna keep looking and will post if I find anything lol