When was Latin language invented?

Latin wasn't "invented" in the way that a language like Esperanto was. It evolved naturally over time from earlier Italic languages, which in turn descended from Proto-Indo-European.

While we don't have a precise date for when Latin *became* Latin, it's generally believed to have emerged around the 8th century BC in the region of Latium, central Italy. This is the time when Latin started to be spoken by the Romans, and its influence spread as the Roman Empire grew.

So, instead of an "invention" date, we can say that Latin emerged around the 8th century BC and flourished for many centuries, becoming the official language of the Roman Empire.