>Before HD, almost all video was non-square pixels

Correct. This came from the ITU-R BT.601 standard, one of the first digital video standards authors of which chose to define digital video as a sampled analog signal. As analog video never had a concept of pixels and operated on lines instead, the rate at which you could sample it could be arbitrary, and affected only the horizontal resolution. The rate chosen by BT.601 was 13.5 MHz, which resulted in a 10/11 pixel aspect ratio for 4:3 NTSC video and 59/54 for 4:3 PAL.

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?

Yes I think they are conflating square pixels with square pixel aspect ratios.

If a video file only stores a singular color value for each pixel, why does it care what shape the pixel is in when it's displayed? It would be filled in with the single color value regardless.