This is a great but very hard question. Thank you so much! I'm a Classicist and this gave me a cool opportunity to read about Early Modern mail systems. Made my weekend!

The short answer, I think is no. But there are nuances. All of the resources I could find on house numbering systems say the first direct evidence of a house-numbering system comes from the Early Modern period and that they were most common in France and Switzerland. This doesn't prove that systems of this type were invented at then and there, but it suggests that they weren't common until this time.

Coming back into my comfort zone, I can tell you that I don't think that the Romans used a house-number system because all of the Classical evidence suggests that no such system was in place. The main use of these systems in the Early Modern societies which implemented them seems to have been voting registration and mail delivery. But the Roman electoral and mail systems were run completely differently (and much less formally or exactly).

The Republican Roman electoral system was a complicated, unfair, and unobjective mess, which is why it was eventually all but replaced. Pre-Empire, Romans voted in tribes (we're not gonna talk about curiae today, because those are even weirder). Originally, these tribes were supposed to be geographically defined, like the electoral college in the modern US. But, when the Romans started expanding and giving conquered people and ex-slaves citizenship and voting rights, the tribes became a hot-button political issue for the Patricians (who were, to my knowledge) all enrolled in the 4 original tribes. It's very complicated, but the short explanation of what happened is that by about 250 BC, the tribes were heavily gerrymandered to the point where they were completely defined by the whims of whoever was in power. New citizens were added to tribes low-down on the voting agenda (so they had to vote last, but as voting stopped after a certain kind of majority was reached among the 4 original Roman tribes, this functionally meant they never got to vote at all unless a tie-breaker was desperately needed).

The mail system was far more disorganized, to the point that it was almost entirely ad hoc for the average citizen. If you needed a letter delivered to a friend, you gave it to someone who was traveling in that friend's direction. This is why so many Roman letters include an introduction for the delivery guy (see Augustine's introduction here). It also explains why so many Roman letters survived to the present day: because you were constantly worried about a letter not arriving, you always made copies and usually sent 2 or 3 by different couriers to make sure one actually arrived. Because we know that lots of these letters did get lost, I'm assuming that there wasn't a reliable way to locate someone's house even if you were very near it. So I don't think there was a numbering system. If you want to know more about how Roman mail systems and travel worked, ORBIS is a cool tool which shows you how long, expensive, and dangerous it could be to travel around the empire, which maybe shows why Romans only trusted good friends to deliver their letters.

I hope this helps!