So now I spent several hours trying to fix missing MAC address on my old motherboard just to find out that almost everything was done right by me, just the EMS DOS driver was making problems.
The board was an old ASUS M3A-H/HDMI, which still works pretty good, but some years ago the network stopped working in Windows, it got correct IP configuration via DHCP, it could send packages but didn't receive any. On Linux everything worked ok. The problem was that the MAC address disappeared from BIOS when I flashed a new version as I found out today.
So my only solution, while I was using the motherboard on a regular basis, was to manually set the MAC address in Windows to the same as Linux was generating as the one generated by Windows wasn't working with my router. (Of course it still could be the case that Windows wasn't generating any and just showed some bullshit.)
Today I was setting up a new PC with older components and did a research which pointed me to the blog entry of trweb which described my problem and had some thoughts and a solution suggestion. I found the mentioned DMI236.EXE and tried it out by booting FreeDOS via an USB flash drive created with a very useful utility YUMI. I was sad to find out that DMI236 crashed with some cryptic error "Illegal Instruction occurred". The also mentioned AFUDOS.EXE (AMI BIOS flashing utility) didn't bring the solution. ASUS offered only a modified version without MAC setting options and the unmodified AFUDOS by AMI was telling me that my BIOS didn't support it. Hours later trying FreeDOS Live CD, modified Live CD with some other settings and more I stumbled across a thread which told me that the problem was occurring in combination of some DOS programs and EMS implementation of FreeDOS EMM386.EXE providing EMS and the solution was to just use XMS provided by HIMEM.SYS et voilà - DMI236 worked, set the MAC address and both Linux and Windows showed the correct address and worked without any quirks.
All at all it was a productive day and I had some fun going back to the ancient DOS days.