Doesn't this apply to Real Madrid or Bayern? Bayern wins the Bundesliga every year and for them this is a routine/norm that they cannot fall below and every year they want to win the Champions League. Real won the Champions League three times in a row and 4 times in 5 years. It is difficult to imagine that at some point they will say “perhaps we have enough.” Each top team strives to win all tournaments in which it participates. As for personal meetings between them, I hope that City will be knocked out earlier and there simply won’t be any 
Of course it applies to both Real Madrid and Bayern Munich, i just think what City have achieved in recent years is a little bit more special because i think the EPL is more competitive than the Bundesliga and also the La Liga. Bayern and Real Madrid have such a great history that is a dream to City, but i think the English champions are going to get there someday, the only problem is if Pep decides to leave soon, they would not be able to replace him, there is no manager who is better.Having said that, i think now that City have won their first CL and the weight of not winning it has been removed from their shoulders, they are even more dangerous, now they play without any pressure and they can dominate the competition like they are doing in England.
City were champions before Pep and will be champions after Pep (of course, unless the League can punish them for 115 FFP violations and send them two divisions down, but I have little faith in this because only Everton can be punished). There is nothing special about him.
As for the pressure, I think this is just a feature of tournaments where there is a playoff stage - there is a lot of randomness. You can very effectively “buy” a round-robin tournament by flooding everything with money (in the long run your investment will outweigh everything else), but in the playoffs everything is possible.