No, you don't have to explicitly analyze a word from a feminist perspective for that to happen. If you think so, you've fundamentally misunderstood how culture promotes things, as well as the point of analysis. Analysis describes explicitly a concept or fact that was always there, whether it was earlier analyzed or not. Feminism certainly didn't create a culturally created power imbalance between sexes, it described one that had already existed for quite some time, for a relevant example. Your argument seems to be that people who don't think with feminist vocabulary cannot act in ways that vocabulary describes. That's... pretty preposterous.
In the case of Mario, let's take the narrative structure of most games. Saving Princess Peach is your goal. You're going on your quest for her sake, because it's the right thing to do. That's fine. But she doesn't really serve any purpose in the story other than as a goal to be quested after. What's worse, a romantic relationship with her (or a kiss, or whatever) is portrayed as a
reward Mario earns through hard work on her behalf. Making it a transaction is the problem with this, because it implies she's obligated to do something like this for Mario whether or not she wants to -
he's earned it, and that's what matters to the story*.
Now, you don't have to explicitly think any of this, or agree with it, for it to support the nebulous concept of "How stories are supposed to work". And since virtually everyone conceptualizes their life in terms of stories, this in turn contributes to the nebulous idea of how people normally treat each other and what's generally acceptable, without Mario being particularly responsible. It just becomes what's normal, and that's the problem in this case. It's not examined, so it's not rejected, but it isn't outright accepted either. As you say, very few people would say, "Women are property". So they don't. It just sort of lingers, allowing people to
avoid having to think, "This woman is a person, just like me".
*There's a whole rant here about people who do "nice things" because they think it entitles them to sex. But that's a tangent. Important point, though, is that are a lot of people who
do buy into the narrative structure without ever explicitly internalizing the idea that women are property.
EDIT:
Maybe we are just meant to find what ever interpretation frames her in the best light like some sort of zealot who picks the versus they like to take literally and other things mean something else, rather than approaching with a skeptical mind.
I'd really prefer an argument that dealt with her clearly understood meaning, instead of the exact words she used to express it. Sure, she could've been more technically precise. In a Youtube video. One not intended for a scientific or professional audience, for that matter. When we're debating the precise definition of the word, "epidemic", as if it's actually relevant to any claim she's made, we've clearly lost our minds.