This is the part of the game that became a meme for how "cringe" it is. Sure, if by cringe you mean totally hilarious and silly and cool. Amelie calls herself Princess Beach and it's such a powerful joke that it echoes throughout history and sometime in the past i'm sure even a caveman heard it and died laughing. That was the last pure Neanderthal so we all have Kojima to thank for the human race, which i guess might be a bad thing now that i think about it, but fuck you the joke is still funny.
The fact that everybody in this game talks in gamer references is actually a neat little piece of world building that i appreciate. This is a world where everyone is isolated in their own little crappy holes, of course everyone would be a gamer. Take it from someone who spent most of his life isolated in a crappy hole, being a gamer comes with the territory.
So when Higgs is constantly ranting about boss battles and game overs it's not because he's self-aware or breaking the 4th wall, it's because he like everyone in this world is an isolated weirdo who had nothing to do all his life but sit in his little room playing video games. The world informs the characters and in turn their little quirks inform us about the world. It's GOOD.
Amelie IS the princess peach of this journey so of course the characters would reference that, and her little Princess Beach pun followed by the two of them running along all silly and happy as if in celebration of the world's greatest joke being made is in fact high art and the game is a masterpiece. That's my opinion and you'll never take it away from me.