It could be some form of copy protection. I've seen that before, eg in the game F29 Retaliator.
The game will do some low-level access to the floppy to scan for some non-standard stuff. A regular copy program that only looks at the FAT contents, will not be able to copy the disk at the level required.
Sometimes it can be that they specifically 'damage' a certain sector on the disk. So if you copy it to a 'good' disk, the check fails.

It will take some reverse-engineering of the code to see why it concludes that the program 'has not been installed properly!'.

I just tried copy2pc v. 5.20, and v. 6.00 and made a copy. And the result is the same! Shouldn't this have solved it?

I guess the floppy got bit rotten, because I remember that I was also able to install the game 20 years ago, and now this isn't possible anymore, it doesn't install, and you can only run it from the original floppy if you want the VGA version... By the way, I'm not using a much newer computer, it's a Pentium 166 Mhz with cache disabled with setmul.

md5sum between original and copy is exactly the same!

Older games used all kinds of sneaky copy protection. One particular method was a "weak bit" or a "flaky bit" which cannot be duplicated using standard floppy hardware, and sometimes such bits can indeed decay.

I'd suggest that you find someone with a kryoflux. Thats no guarantee but chances are higher to preserve an image that can be written back in a working condition. AFAIK there is no floppy emulator for x86 that emulates copy protected disks (maybe not even an image format).