Insufficiently unique aspects of existence could be 'tacked on' to already-worshipped gods in both Greek and Roman worship - such as Hermes/Mercury being able to be worshipped as the god of, specifically, gaming, herding, or translators! Epithets like "Hermes Kriophoros" ("Hermes the Goat-Holder") or "Hermes Agoraios" ("Hermes of the Marketplace" - used for merchants and thieves!) would invoke the more-specific aspect of the god in question to your problem-of-the-day!

As for ridiculously specific gods, the Romans had quite a few - for one, they had very little compunction about inventing new gods - as gods, to the Romans, represented divine oversight over aspects of reality, it stood to (Roman) reason that any sufficiently unique aspect of existence, even a small one, would have a corresponding deity. We have such titans of divinity as... [checks notes] Terminus (god of boundary markers), Nenia (goddess of mourning), and Cardia (goddess of door hinges).