The terms "early", "mid" and "late" game are very loose terms to give a general idea of timing in the game. Therefore it is difficult to connect them with certain techs, as everyone uses different techorders to get ahead and i.e. galaxy size and tech speed also has an impact.
That said, here's my approximation:
"early" game = during the colonization wave and the second colonization wave (the extreme planets), warships are small hulls in small fleets with weapons up to harpoon level. War at this point is an annoyance, hardly a direct threat to survival.
"mid" game = after the first colonization wave and usually after the second, medium hulled ships are starting to enter combat, "harpoon" level weapon tech is standard, defenses come into play on ships. War at this point can be very painful and enemies with planetary invasion can get nasty.
"late" game = when you've reached the end of a few tech branches and are far along the weapon, invasion and hull branch. Everything is big, your ships, your planets, your weapons, your economy, etc... War at this point means big battles and invasions till the enemy is dead. The end is in sight, either through conquest, tech victory or other victories.
That said: victory can be achieved under conditions that would otherwise be known as "early" or "mid" game. It is usually a rough approximation of tech level and empire development.
I usually average about 5 years on a game, but recently I played one that took 11 years to complete, so time is really no indicator. You should just be having fun, it's not a race!