She's gonna be sore tomorrow

It's not a video game reference, プレイ is commonly used this way.

The balloon in the bottom right said something completely different.

Another case of being too literal (or oddly, less literal in this case, since many direct references were ignored). She says "What sort of PLAY shall we try today". That is not the same as saying "How shall we do it today". She then later says "FIVE PLAYER?" which is not the same as "five way?" It's all based around Rise's character and her view of everything as just being a "game" or a "bit of fun". Another example of the fact that no matter what you know about Japanese, it's not worth much if you don't know anything about the character's you're translating.

5P is very literally a 5-way sex act, regardless of context. And as memento mori already said, the English word 'play' is normally used to describe a variety of (fetish) sex acts. There's nothing out of the ordinary with the word choice and it has nothing to do with the English usage of 'play' in reference to games.