Rematches are limited to three games, or at least that used to be the case, and that's per day. So unless you're meeting a player more than that, you should be fine.
Problems with aborting games and the league system

I guess then I am indeed getting the same opponents several times a day, since it is very common for me to notice that I am receiving no league points. Of course there are other requirements to receive league points, which I don't necessarily remember, so it is possible that my games don't satisfy those requirements.

I guess then I am indeed getting the same opponents several times a day, since it is very common for me to notice that I am receiving no league points. Of course there are other requirements to receive league points, which I don't necessarily remember, so it is possible that my games don't satisfy those requirements.
Just because the end game dialogue doesn't show trophies doesn't mean they weren't awarded. There is some code that checks that at game end and if the code doesn't receive the acknowledgement quicky enough, it will just show the game didn't receive trophies. In that case, the trophies are added to your total.
I am trying to get to higher leagues but I have encountered an interesting issue.
The league points are earned by winning games, however, rematches don't count. This would not be a problem in theory, since one does not need to accept rematches. However, it is possible that you are paired with same opponent by the matching system right after your game (or having few games in between), and the league point system considers this kind of game a rematch, hence giving no points. This is super annoying, since I don't want to play a game where I cannot earn league points, however, chess.com has this weird policy of not liking aborting games. Thus I am afraid that I am eventually going to placed in some rude-player-pool or something like that, since I just want to avoid playing the same opponent again.
Finally, you may think that this kind of pairing is not very common and thus not an issue. This may be a case at the lower level where there are more players, however, the higher elo players have, the less same rated players are at the moment playing, and thus there is higher change of non-intentional rematches.
I think this problem of unwanted rematches should be addressed and not only because of the league system, though that is the biggest issue.