Alright, there are a couple of questions and a lot of factors here. Resolutions are a really interesting field of psychooptics meeting marketing.
First of all, why are the vertical resolutions on youtube multiples of 360. This is of course just arbitrary, there is no real reason this is the case. The reason is that resolution here is not the limiting factor for Youtube videos - bandwidth is. Youtube has to re-encode every video that is uploaded a couple of times, and tries to use as little re-encoding formats/bitrates/resolutions as possible to cover all the different use cases. For low-res mobile devices they have 360x240, for higher res mobile there's 480p, and for the computer crowd there is 360p for 2xISDN/multiuser landlines, 720p for DSL and 1080p for higher speed internet. For a while there were some other codecs than h.264, but these are slowly being phased out with h.264 having essentially 'won' the format war and all computers being outfitted with hardware codecs for this.
Now, there is some interesting psychooptics going on as well. As I said: resolution isn't everything. 720p with really strong compression can and will look worse than 240p at a very high bitrate. But on the other side of the spectrum: throwing more bits at a certain resolution doesn't magically make it better beyond some point. There is an optimum here, which of course depends on both resolution and codec. In general: the optimal bitrate is actually proportional to the resolution.
So the next question is: what kind of resolution steps make sense? Apparently, people need about a 2x increase in resolution to really see (and prefer) a marked difference. Anything less than that and many people will simply not bother with the higher bitrates, they'd rather use their bandwidth for other stuff. This has been researched quite a long time ago and is the big reason why we went from 720x576 (415kpix) to 1280x720 (922kpix), and then again from 1280x720 to 1920x1080 (2MP). Stuff in between is not a viable optimization target. And again, 1440P is about 3.7MP, another ~2x increase over HD. You will see a difference there. 4K is the next step after that.
Next up is that magical number of 360 vertical pixels. Actually, the magic number is 120 or 128. All resolutions are some kind of multiple of 120 pixels nowadays, back in the day they used to be multiples of 128. This is something that just grew out of LCD panel industry. LCD panels use what are called line drivers, little chips that sit on the sides of your LCD screen that control how bright each subpixel is. Because historically, for reasons I don't really know for sure, probably memory constraints, these multiple-of-128 or multiple-of-120 resolutions already existed, the industry standard line drivers became drivers with 360 line outputs (1 per subpixel). If you would tear down your 1920x1080 screen, I would be putting money on there being 16 line drivers on the top/bottim and 9 on one of the sides. Oh hey, that's 16:9. Guess how obvious that resolution choice was back when 16:9 was 'invented'.
Then there's the issue of aspect ratio. This is really a completely different field of psychology, but it boils down to: historically, people have believed and measured that we have a sort of wide-screen view of the world. Naturally, people believed that the most natural representation of data on a screen would be in a wide-screen view, and this is where the great anamorphic revolution of the '60s came from when films were shot in ever wider aspect ratios.
Since then, this kind of knowledge has been refined and mostly debunked. Yes, we do have a wide-angle view, but the area where we can actually see sharply - the center of our vision - is fairly round. Slightly elliptical and squashed, but not really more than about 4:3 or 3:2. So for detailed viewing, for instance for reading text on a screen, you can utilize most of your detail vision by employing an almost-square screen, a bit like the screens up to the mid-2000s.
However, again this is not how marketing took it. Computers in ye olden days were used mostly for productivity and detailed work, but as they commoditized and as the computer as media consumption device evolved, people didn't necessarily use their computer for work most of the time. They used it to watch media content: movies, television series and photos. And for that kind of viewing, you get the most 'immersion factor' if the screen fills as much of your vision (including your peripheral vision) as possible. Which means widescreen.
But there's more marketing still. When detail work was still an important factor, people cared about resolution. As many pixels as possible on the screen. SGI was selling almost-4K CRTs! The most optimal way to get the maximum amount of pixels out of a glass substrate is to cut it as square as possible. 1:1 or 4:3 screens have the most pixels per diagonal inch. But with displays becoming more consumery, inch-size became more important, not amount of pixels. And this is a completely different optimization target. To get the most diagonal inches out of a substrate, you want to make the screen as wide as possible. First we got 16:10, then 16:9 and there have been moderately successful panel manufacturers making 22:9 and 2:1 screens (like Philips). Even though pixel density and absolute resolution went down for a couple of years, inch-sizes went up and that's what sold. Why buy a 19" 1280x1024 when you can buy a 21" 1366x768? Eh...
I think that about covers all the major aspects here. There's more of course; bandwidth limits of HDMI, DVI, DP and of course VGA played a role, and if you go back to the pre-2000s, graphics memory, in-computer bandwdith and simply the limits of commercially available RAMDACs played an important role. But for today's considerations, this is about all you need to know.