ZDoom

For some reason, GZDoom is registering the frames of an animated texture as 1 single texture. And when I try to launch the game, it thinks that the other texture in the animdefs range don't exist.

the error is telling you the problem. COMPSRVR4 doesn't exist. It's got a different name I presume by your screenshots.

It could be because your file names are 9 characters long.

Doom was a DOS game and, at the time, 8 characters in a file name was the limit. Perhaps that's why Doom internal lumps have a max of 8 characters?
Anyway, the default is still 8 characters. You can get around this in many cases by providing full path names in your control files but, wherever possible, it's usually easier to stick to 8 character (or lower) names.

So, instead of COMPSRVR1, try using file names like COMPSRV1 or something.