I have found the solution.
First of all: The error message EGL library doesn't support emulator extensions does not come from the eglfs_kms plugin itself, but from the emulator plugin eglfs_emu which is used as a fallback, since the original plugin could not be loaded.
So why could eglfs_kms not be loaded in the first place?
The answer is that the library, resp. symlink libQt5EglFsKmsSupport.so.5 had not been deployed on my target device; therefore, loading the eglfs_kms plugin failed. Deploying the library with the appropriate symlink fixed the problem, i.e. the filesystem should look like this:
~ # ls -l /usr/lib/libQt5EglFsKmsSupport.so.5*
lrwxrwxrwx ... /usr/lib/libQt5EglFsKmsSupport.so.5 -> libQt5EglFsKmsSupport.so.5.12.8
-rwxr-xr-x ... /usr/lib/libQt5EglFsKmsSupport.so.5.12.8
I found this out by doing an strace on one of Qt's example applications:
export QT_QPA_EGLFS_INTEGRATION=eglfs_kms
strace /usr/lib/qt5/examples/widgets/tutorials/notepad/notepad 2>&1 | tee log.txt
and then greping for failed open-statements.
@Pablo-J-Rogina Thanks for your time!