I'm confused... what does DVD, SD or any arbitrary frame size have to do with the shape of pixels themselves? Is that not only relevant to the display itself and not the file format/container/codec?
My understanding is that televisions would mostly have square/rectangular pixels, while computer monitors often had circular pixels.
Or are you perhaps referring to pixel aspect ratios instead?
It comes about from digitizing analog video signals. The early standards for sampling (digitizing) analog video signals resulted in the digital pixel horizontal sample size (often) being wider than the line spacing of the displayed analog video. With the result that digitized video of analog signals usually has a "pixel size" (analog video has no concept of discrete horizontal pixels) that is wider than it is tall.
CRTs didn't have pixels at all. They had shadow masks (or aperture grilles) and phosphors, which could be a triad of rectangles, lines spanning basically the entire screen height, or dots. They did not line up with the signal, so it doesn't make sense to call them pixels.
A square pixel has a 1:1 aspect ratio (width is the same as the height). Any other rectangular pixel with widths different than their heights would be considered "non-square".
F.ex. in case of a "4:3 720x480" frame… a quick test: 720/4=180 and 480/3=160… 180 vs. 160… different results… which means the pixels for this frame are not square, just rectangular. Alternatively 720/480 vs. 4/3 works too, of course.
CRTs do not have pixels. At all. The shapes you might see on the screen if you look really closely at it are solely different bands of color phosphors. CRTs are capable of drawing arbitrary beam shapes on them in one color[0]; but you need regularly spaced filters and phosphor patterns in order to get multiple colors. If these """pixels""" were bigger, you'd see a perfectly normal red part of the image, next to a perfectly normal green part of the image, next to a perfectly normal blue part of the image.
What a CRT actually draws, though, are lines. Analog television is a machine that chops up a 2D plane into a stack of lines, which are captured, broadcasted, and drawn to the screen with varying intensity. Digital television - and, for that matter, any sort of computer display - absolutely does need that line to be divided into timesteps, which become our pixels. But when that gets displayed back on a CRT, the "pixels" stop mattering.
In the domain of analog television, the only property of the video that's actually structural to the signal are the vertical and horizontal blanking frequencies - how many frames and lines are sent per second. The display's shape is implicit[1], you just have to send 480 lines, and then those lines get stretched to fit the width[2] of the screen. A digital signal being converted to analog can be anything horizontally. A 400x480 and a 720x480 picture will both be 4:3 when you display it on a 4:3 CRT.
Pixel aspect ratio (PAR) is how the digital world accounts for the gap between pixels and lines. The more pixels you send per line, the thinner the pixels get. If you send exactly as many horizontal pixels as the line count times the display's aspect ratio, you get square pixels. For a 4:[3] monitor, that's 640 pixels, or 640x480. Note that that's neither the DVD nor the SD cable standard - so both had non-square pixels.
Note that there is a limit to how many dots you can send. But this is a maximum - a limitation of the quality of the analog electronics and the amount of bandwidth available to the system. DVD and SD cable are different sizes from each other, but they both will display just fine even on an incredibly low-TVL[4] blurry mess of a 60s CRT.
[0] There were some specialty tubes that could do "penetrative color", i.e. increasing the amplitude of the electron gun beyond a certain voltage value would change to a different color. This did not catch on.
[1] As well as how many lines get discarded during vertical blanking, how big the overscan is, etc.
[2] Nothing physical would stop you from making a CRT that scans the other way, but AFAIK no such thing exists. Even arcade cabinets with portrait (tate) monitors were still scanning by the long side of the display.
[3] There's a standard for analog video transmission from 16:9 security cameras that have 1:1 pixel aspect ratio - i.e. more pixels per line. It's called 960H, because it sends... 960 horizontal pixels per line.
https://videos.cctvcamerapros.com/surveillance-systems/what-...
[4] Television lines - i.e. how many horizontal lines can the CRT display correctly? Yes, this terminology is VERY CONFUSING and I don't like it. Also, it's measured differently from horizontal pixels.
I'm not 100% sure I understand your question, but in order to display a DVD correctly, you need to either display the pixels stored in the video stream wider than they are tall (for widescreen), or narrower than they are tall (for 4:3). Displaying those pixels 1:1 on a display with square pixels would never be correct for DVD video.