Is there any bus/coach travelling direct from Brussels, Belgium to Tbilisi, Georgia?

If such exists, they are a well kept secret. You'll likely have to do this in stages.

The most attractive option I could find (excluding flights) would be to take trains as far as Istanbul and then there is a bus from Istanbul to Tbilisi.

The train segment is about 50 hours (including three nights where you could opt for comfortable sleeping compartments and a ten hour, daytime, stopover in Sofia, Bulgaria). The bus is, reportedly, another 36 hours (so at least one night on the bus). All told, this is about a 4-4.5 day long trek depending on how long you need to wait in Istanbul.

You can probably patch together a series of bus trips to get to Istanbul (or via Ukraine if you don't mind a ferry ride across the black sea) if rail travel is not an option but they are likely to be slower and unlikely to be cheaper. And they are certainly going to be less comfortable.

To put it into perspective, a flight (with a stopover in Istanbul) will take about 12 hours and cost under 200 euro (depending on date of travel). I assume, however, that flying is not an option for you for some reason.

Nope, but you can travel:

1. Brussels-Munich on Eurolines or Flixbus
2. Munich-Istanbul on Alpar Turizm, Ulusoy or Yazicioglu Turizm
3. Istanbul-Tbilisi on various carriers, such as Metro Turizm.