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You're misunderstanding. Almost all major DOS games right up until about the mid-90s used a single common screen resolution of 320x200. When improvements in display tech allowed for more capable graphics cards, the priority was always for more color depth, not higher resolution. So the large majority of games were 320x200@2bpp (4 colors) with CGA, then 320x200@4bpp (16 colors) with EGA, then 320x200@8bpp (256 colors) for VGA. [1]

CRTs themselves have no concept of pixels or pixel density -- they're analogue devices where the primary limiting factor is the refresh rates supported by the hardware -- so the screen resolution in pixels is entirely a function of the display hardware generating the video. Almost all CRTs were 4:3, so typically you'd have a 320x200 game being displayed at a physical 4:3 resolution on the monitor. This means that in most cases, the pixels were not square, but rather each pixel was slightly portrait-orientation, having its own aspect ratio of 5:6. Most games artists took this into account, so if they wanted to draw a square that was 30 pixels across, they'd make it 25 pixels tall.

So the point of the 1600x1200 number is that this is the lowest resolution at which a display with square pixels can replicate the exact aspect ratio of both the screen and the pixels used in those classic games. Note that the aspect ratio of the pixels is 5:6 -- 5 is a prime number, so you can't reduce the ratio further. So if you turn each pixel in the original 320x200 screen into a grid of 5 by 6 square pixels, you get 1600x1200.

So only 1600x1200 (or an exact multiple) can provide a pixel-perfect rendering of the graphics from games from that era. Add a filter that emulates some of the visual artifacts from CRTs of the era -- scanlines, bloom, etc. -- and you're getting a pretty exact recreation of the experience you'd have had playing games in the DOS era. [2]

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[1] SVGA games at higher resolutions started becoming more common around 1994/95, but within a couple of years after that, Windows began to displace DOS as the primary target platform for PC games, leaving the vast majority of DOS games in the 320x200 camp. There were some outliers that used e.g. EGA's 640x350 mode, or otherwise supported higher resolutions at the expense of color depth (SimCity is a major example that comes to mind, along with a few games that used 320x200 but expected square pixels (producing a 16:10 screen aspect) but these are exceptions rather than the rule.

[2] At least visually. There are a lot of other sensory elements of classic gaming that remain entirely absent with modern hardware -- the static charge on the glass of the CRT monitor, the smell of air passing over the hot electronics in the PC, the mechanical sounds of reading data from floppy and hard disks, etc.