So I have noticed that I am doing far beyond 20 reps of push ups and still, after I have done about 50 of them, or even 2 sets of 25 my arms feel like they're expanding and growing muscle, both the biceps and triceps. This is confusing to me because I thought that over 20 reps only builds stamina and endurance, not stronger muscles. If this is the case, why do my arms feel like they're growing? Are they?
You're right that certain rep ranges primarily cause certain types of adaptation. But that's not the end of the story. A good example of that is the legs of high level cyclists. Most of these folks aren't doing strength training, and certainly very few are training for hypertrophy, but their legs are both strong and tend to be very muscular.
In the short term, that "growing" feeling you're experiencing is the pump. From exrx:
Immediately following a weight training exercise, the muscle may seem full and tight for 15 to 30 minutes, or 'pumped'. The muscular 'pump' is caused by trapped plasma within the muscle.