The point isn't to build categories for their own sake, the point is that categories allow us to group and exclude concepts based on whether they have similar "rules" or "feels". If you're designing a new thing or analyzing an old thing, it's useful to be able to sort through which heuristics and rules will likely apply to the thing you're examining.

In other words, if we have generally good advice about how to make a game fun, and you know that advice is predicated on a certain definition of "game", it is easier to figure out whether or not that advice applies to the thing you're building.

Rosewater's categorization of Minecraft as a toy rather than a game is a really good example:

If you're a game designer, and you think of Minecraft as a game, there are a lot of game rules that Minecraft breaks. Its progression system is wack. It has a lot of design elements that don't really feed into its core loop well. And part of understanding why those "flaws" don't matter is understanding that Minecraft is working under a completely different set of design rules than a game like Doom.

So one of the things we do as designers is we add distinctions between different types of experiences that allow us to make rules that are more consistent and predictable. If you're able to split computer experiences into different categories and say, "Minecraft and Doom are fundamentally different", then it's easier to come up with rules around flow, and progression systems, and difficulty curves, and unity of theme.

As to why we need those rules in the first place? Because we want to make good games. Without narrowing a design space or coming up with heuristics to filter out good and bad ideas, designing games is just too stinking hard. There are always going to be exceptions to those rules, and that's fine. But the goal is still to give us mechanisms that in most cases allow us to talk about why an idea does or doesn't work.

This is the same reason why games build player archetypes. Basically no one fits into a single player archetype, and there's heavy overlap between them, and there are players that defy all of the archetypes we come up with. That does not mean that player archetypes aren't extremely useful as design tools. People think about classification systems like they're some kind of moral or philosophical separation between objects. They're not, classification systems are entirely pragmatic, practical tools.

All categorizations are wrong. No maps are completely accurate. But game design is an extremely fuzzy field, and it's hard to figure out what makes "good" design when your subject is so broad that literally every rule you come up with has exceptions. As a designer, I have multiple classification systems that I use for games and players, and some of them outright contradict each other. But all of them are useful lenses that I look through when I need to think about design decisions.