What does this mean? Does soccer not have a single winner take all championship at the end of the season like The Super Bowl or NBA championship?
These are the final winners in each competition ...
The premier league is a season long league. All 20 teams play each other twice (home and away) with 3 points for a win and 1 for a draw. Whoever has highest points tally, or best goal difference if equal, wins. Unlike US sports, the bottom teams get relegated and the top teams of the lower league get promoted. It's possible to drop down more than one tier, City once got relegated, then relegated again (so we were in the 3rd tier of English football!)
The FA cup is a knockout competition open to all league clubs (there are 4 leagues in the UK*). The larger clubs tend to win, but not always - some non-league teams have good runs, sometimes there are some shock "giant killer" games where a big club gets clobbered by a lower league one.
The champions league is where the top handful of clubs from each national league compete in a group stage initially, then a knockout competition for the last 16 (same format as the world cup).
There is a new club world cup, that is something else City could potentially win this season.
Why winning the treble is such an achievement is the number of matches played if you're in all these extra cup / knockout competitions. Imagine a team playing in an extra NFL league in the same year ...
* fun fact: Manchester United were the only previous English club to win the treble back in 1999/2000 ... at the time City were just clawing their way out of the 3rd tier league, which is how far we'd fallen. No one would have imagined then that City would be the ones to equal it!