Bayonetta is a video game, and evangelion is a manga-turned-anime. I was completely unaware that either one had songs called "fly me to the moon" though.
Evangelion was an anime first. Each episode ended with one of the voice actors singing Frank Sinatra's "Fly Me To The Moon." Some of them... definitely do not speak English.
English with Japanese accent is really funny IMO. Until you start learning Japanese that is, and a lot of words are imported from English... badly. I'm not a native English speaker but my English understanding is pretty good, and even so, katakana English can be very hard to understand without and even with transcripts.
For Evangelion it was the closing Theme for the anime episodes. As it is such an iconic feature, it was reused (and reimagined) within the movies (the new ones)