Why is wide screen a thing in Diablo (Classic) but not in the GOG Diablo version? Shouldn't it be the other way around?

(Side note: I can't tell the difference between textures in both versions. They both look like the same to me)

Diablo 1 was created and patched to 1.09 before widescreen was ever a thing. The DX version only has it, simply because it's not the classic version of diablo and it's been added in, ofcourse with none of the work to make it look like it's actually widescreened, it's mostly just stretched.

The non-classic version looks like it isn't widescreen, but if you check your monitor, you'll see that you're actually playing in full HD. There is also no stretching to fit the different aspect ratio, just big black bars in the side. This version uses the same textures so there's no high resolution benefit in terms of quality, but at least your monitor is not having resolution fits.

The classic version forces your monitor into classic resolution and that likely has the image stretched. It will look widescreen'ish, of course, but those old 4:3 graphics pushed into 16:9 or 3:2 takes some getting used to. And you're not really gaining any additional screen space by having the 4:3 ratio stretched out instead of having black bars inserted.

Classic doesn't have widescreen. If it did everything would would be too small. Playing classic version just means everything is stretched out.

It's not flipped. The issue is your understanding of widescreen. Games can be widescreen and totally run in a small window. Widescreen doesn't automatically mean "fills up my whole display". Classic version doesn't keep aspect ratio, so it fills up your whole screen.
The "improved" version keeps aspect ratio, so you have black boxes around the game window, but the actual resolution of the game window is bigger and widescreen.

To make the "improved" Diablo fill more of your screen, you have to set up the display mode to the nearest 4:3 resolution to your actual display resolution and use fullscreen or windowed mode, without integer scaling.

It's not that classic 'supports' widescreen. It's that the original game was made before widescreen monitors were a standard, so it doesn't know how to properly handle itself on them. It just fills top left corner to bottom right corner, no matter what.

If you were playing on an oddball super tall and thin monitor, classic mode would just be stretched up and down instead.

The enhanced version is coded to understand screen ratio and to properly keep it's shape. The Classic version isn't messed with at all to maintain Battle.net compatibility.