That’s because there aren’t any, at least not anymore. See this thread on Watchuseek for a discussion on the matter.
The only names you’re going to find associated with the area (i.e. the Balkans) are for vintage/resurrected brands such as Darwil (Swiss, but had a facility in Trieste so in the Italian Balkans), clock (not watch) companies such as Insa (which mostly do gauges and dials now), and defunct brands such as Mestis (which appears to have used ebauches from Poljot and 1MWF).
Basically keep it in storage because of the thought behind it. If the gifter asks where it is, you can always say you’re storing it because you don’t want to risk losing it/having it stolen.
Over time/distance, you can get rid of it, use it for parts (my preferred method), or find some other means of disposal (e.g. target practice). If asked about it, depending on who’s asking, you can say you lost it, or that it stopped working and was too expensive to repair so you sold it, that you had it customized (if you kept it and removed the unauthorized branding), and so on. If you believe the gifter can handle being told the brutal truth (or this is one of several gifts you’ve received from that person so one of them being faulty/fake is okay) perhaps at that time you can tell the person the plain truth.
As for fake watch disposal, for me it depends on what it’s made of, what’s inside, and where the unauthorized branding is.