Does the muscle failure of a compound exercise like push ups and pull ups are related to one muscle or a group of muscle?
Firstly, I want to give you an idea about two types of muscle failure. One is induced failure, the other one is unintentional failure.
Why would anyone want to induce failure, you might ask. It's not actually taxing the muscles to the point of making them unusable. You might have heard of the term 'reps to failure'. You do the reps till your limit and push more till you feel like losing control of further reps in that particular session. That's reaching failure. This is done intentionally sometimes to extract maximum work out of the muscles albeit with the inherent risk of an uncontrolled weight on top of you.
Alright, coming to the other type, this is usually causes by unconditioned joints and weak tendons. This is jot a desirable effect, does not help with muscle gain, does not give you any gains whatsoever. And it is avoidable and should be avoided at all costs. It is simple though but I won't go into those details.
Now, back to your question, the muscles that fail are not the whole group involved. It's like what happens in any system that is subjected to stress. The weak point fails first. This is true regardless of whether it is intended or not. Your weakest muscle loses control first. This will be sudden, unexpected and can lead to problems if you don't condition your joints first and you don't want that situation.
Hope this informs you to keep your joints in good condition.