This answer works for all contexts (even outside gaming):
Choking happens when you get nervous, or overly excited. This is common in the exact situation you described in rythm games, but also in plenty of other contexts. For instances, when playing a game like League of Legends, is common for your clicking ability and attention to detail to be heavily reduced when you decide to commit to a fight. I think rocket jumping offensively as Tristana effectively makes me drunk.
I have found that the solution to this is mostly psychological. It might seem overly simplistic, but you only get nervous because you're heavily emotionally invested in the outcome.
There are two ways to stop being nervous about an outcome. The first is to have complete trust in your thought process. In the LoL example, if you've jumped in enough times in a similar situation, you'll eventually know all the outcomes, and therefore you'll remain as calm as in any other situation. In the case of Rythm games this is the reason why after you 100% a song, repeating the score is almost trivial.
The second, and perhaps more useful is to develop a secondary goal to winning/100%ing a song. In the case of a MOBA it can be CS'ing or buying pinkwards. Or even not dying to ganks in X amount of games. In the case of a Rythm game it could be "have less than X of a type of hit or worse in the next Y rounds."
This has two advantages: The secondary goal is less advertised in game and often requires a spread sheet or a calculator after the game to assess. This means that during the game you won't be thinking about it at all. If you miss a note you're probably still alright.
The second advantage, and in most cases the biggest, is that this secondary goals will be much better for real improvement than the one time goal post of winning a game, getting a lot of kills or 100%ing a song for the first time. If you only miss 20 notes over 20 rounds you're probably a lot better than the guy that misses 50 but gets the song perfect once (unless you're always missing the same note, then you need to get that shit right.)
Of course the hard part in this mental trick is getting yourself to trully value the secondary goal more than the first, obvious goal that you've had for a longer time. Feynman said we're the easiest ones to fool, but not in this context unfortunately. In the context of single player rythm games this might be slightly easier. But when it comes to multiplayer it's hard to overcome the social pressure. After all, everyone else is only worried about winning the game, not whatever secondary goal you set for yourself. Practice is the only solution. If you set secondary goals and go through all the trouble of keeping count, you'll soon find that your brain hates doing chores for no reason, and thus will assume the chore is important.
By the way, this mental trick also works for stage jitters. Simply reframing a talk from a presentation you must nail to some secondary goal like "teaching the audience about something." or "correctly portraying the story of this character" can have a dramatic effect on your nerves.