So the parallel here, if I'm reading your example right, is that even if there was no sexism involved in choosing a woman to be in need of rescuing, it's still sexist to have used it because it happened repeatedly (which I believe MC Final Sigma also put forth). If that's the case, then, again, I'm not seeing eye to eye with that assertion. It's putting sexism where none was intended or implied, and pointing at someone while saying "YOU'RE A BAD PERSON!" when no ill intent was present.
Now, I know what at least a few are thinking. You're thinking, "It doesn't make them a bad person, it just says what they did was bad," or something similar. But there's no positive or gentle spin to put on the term "sexism" when it's thrown out there, as its stigma carries a lot of injustice and anger with it due to the history behind it. So the accusation still puts them in a rather bad light unfairly, kind of like how someone accused of rape can be proven to be 100% innocent, but the stigma of that accusation follows them anyway. As such, it's not a word to be thrown about lightly, and to me, calling the "damsel" trope sexist is really stretching what falls under that term.
Here's the problem; Peach wasn't Mario's. The story goes, Koopas invaded, turned the Mushroom Kingdom people into various objects using black magic, and Peach is the only one who can undo the spell that was used. She's captured by Bowser, and Mario goes off to try and save her.
There's no "she's mine" involved on the hero's part, and Bowser likely locked her away to stop her (it's implied that she can't undo the magic as a prisoner, but not flat out stated). So you're analogy (and MC Final Sigma's) doesn't work to me, because the story is in place, even if it's only in the manual. The gameplay may not differ, but the reasoning behind the gameplay does, and that makes a world of difference... again, to me.