It’s often exaggerated (as you say, bathing was popular in the Roman world, and also in the medieval Islamic world and even in many parts of medieval Europe), but ultimately they did not have sanitation (as in sewers, etc) to the same extent we do, and they had beasts of burden.
That said, I’m not sure to what extent this applies to Ancient Rome (as in Republic to fall of western empire). They _did_ have sewers, with public toilets attached to them, and they probably didn’t have that many horses wandering the street, as the horse collar hadn’t been invented yet. Ancient Rome presumably wasn’t as bad as a large medieval city (which wouldn’t usually have had proper sewers), and things may even have gotten worse later on, because at a certain point bathing really did become a lot less common in much of Europe (due to weird medical beliefs and religious objection to public bathhouses).