8k is double the vertical and horizontal resolution of 4k. You can argue that 4k was deceptive, since it switched from measuring the vertical resolution (2160p) to the horizontal, but 8k is just sticking with the established standards.
4k was the marketers cashing in a puffery token they'd been carrying for quite a long time by accurately measuring the short side of displays.
They moved to measuring the long side AND exaggerating by rounding up in the same generation, stealing the name of a slightly-better existing standard in the process.
You're right that 8k makes sense so long as you accept 4k, though.
Well, 4K is called that because when 1080p became common at home, cinemas switched to DCI 4K and suddenly TV manufacturers had to compete with that, so they branded the closest they could get to DCI 4K as their own "UHD 4K".
Originally 4K came from the DCI spec for digital cinema and it was 4096x2160. For consumer use it was shrinked to UHD which is the 3840x2160, but cinemas still use 4K (although most of the screens actually use 2K)
Now 4K refers to image roughly the width of 4000 pixels according to wikipedia [1]
All new/upgraded auditoriums installed 4K (laser) projectors when I left the business 5 years ago. So not sure if most auditoriums only have 2K these days?
IMAX has only started switching to 4K laser projectors a few years ago, they were one of the last hold outs for 2K projectors (and many IMAX cinemas still use 2K projectors sadly).
Cinema projectors. It should be said that I worked in the business in Norway which had converted all auditoriums to digital in 2010, so many theatres were looking at upgrading their projectors around the time I left for something else.
