So I finely got my first Cultural victory (at least I thought it did) but the game has not ended nor given me any notification that I have achieved.

I have three cities with Legendary culture ranging from 120,000 to 58,000.

I thought that was all that I needed.

What else am i missing?

3 Cities need to meet the requirement (Marathon is 150,000 I think)

That's all 3 making it above the requirement. There is no range, they all have to achieve "legendary culture" which is the number I gave above for marathon

It tells you when a city reaches legendary culture, even if it isn't owned by you.

Are you sure that cultural victory is enabled ?