Beat Saber is absolutely a rhythm game. It doesn't have the typical scoring system that is based on having an accurate timing, but it is far from the only one.
Guitar Hero, one of the best known rhythm game doesn't score based on timing accuracy. You hit a note or you don't, that's all, ut you still have to hit that note with a certain timing. Same thing for Beat Saber, you hit or you don't hit the note. If you want a high score, which is completely optional, as may just be interested in pass harder and harder songs, then there are other mechanisms at play. Guitar Hero has star power, Beat Saber has motion. That it is not pure timing doesn't make them less of rhythm games.
The article says that instead of being a rhythm game, Beat Saber is a "motion game" because it rewards precise and intentional motion. But it is the case for all rhythm games. For games about pushing buttons, the motion is rather basic, but take DDR for instance, even if it rewards timing, the key is how you move on the dance pad. Even if you hit the right arrows, bad movement will mess up your timing and drain your stamina. Even some pure button pushing game like, say, Jubeat is all about making the right movements.
The reason Beat Saber used motion for scoring instead of timing is, I think, that the smooth motion of the VR controllers didn't lend itself well to it. I mean, how to even judge timing? So instead, it judged motion, but it is not just a "motion game" like, say, Just Dance (which is also considered a rhythm game btw).
As for why is it so fun? There is no single thing. It took everything that worked, combined them, executed them well, and boom, a hit.
- Rhythm games are fun, you already have motion controls, motion controls + rhythm is dance, dancing is fun.
- Sabers just work in VR, there are plenty of VR games that feature some kind of sword.
- It is an active game, so it attracted those who want to exercise.
- It has an active modding community, and I thing the devs are smart about it. They tolerate it, without supporting it officially, so modders have a lot of freedom. Most of it is about the sharing of songs without having the right to do so after all.
- It is intuitive with a good learning curve. Most people just need a couple of songs to get comfortable on easy/medium difficulty, but the skill ceiling is high. For example, there is no special symbol to learn, just colors and directions, and on easier levels, they follow naturally (ex: blocks meant to be cut by the left saber are on the left).
- It is a VR game with no locomotion, it limits the risk of motion sickness.
Taken individually, none of these are special, but that's the combination that's great.
Edit: Another argument for "Beat Saber is a rhythm game" is that timing does matter. Unlike in most rhythm games, the timing window decreases as the notes/blocks move faster, simply because you have less time to cut them. Higher difficulties have the blocks move faster, making timing more critical.