I'm always skeptical of people saying "finally a question I can answer", but this is finally a question I can answer with my Eastern European Studies degree.

In a communist state like the USSR, enterprises were generally state-owned. This was after all one of the fundamental principles of communist economics: property and capital was not to be owned by individuals (capitalists - traditionally not a term of ideology, but of social class), but rather by all people manifested in the state. This was not absolute and also differed between communist countries, so in some places smaller firms or argicultural enterprises could actually be private-owned, or collectively owned, but those were more exceptions than the rule: in general, enterprises were part of the state.

It is also worth noting that, despite being state-owned, these enterprises were not the same as government institutions: in many ways they functioned as companies as was known outside of the Eastern Bloc, in terms of governance and structure. The main difference was that there was no individual "owner", but that the owner was a ministry or other government agency. This system sounded nice on paper but was generally completely ineffective in practice. Don't forget that Tetris wasn't an invention by ELORG, they just happened to own the license.

The collapse of the USSR coincided with the fall of communism. This means that the USSR - now succeeded by independent republics including Russia - transitioned to a market economy akin to most of the rest of the world. One of the most fundamental changes in the economic system is the transfer of ownership from the state to private hands: privatization (this takes place on a smaller scale in non-communist countries as well).

Now think about how to transfer virtually every single company in Russia, including shops, restaurants, factories, mines, airlines, from the state to private individuals: how do you even start? Who can get ownership? It's a massive operation. Opening an auction per company for everyone in the country to bid on (like we do nowadays if a single company is privatized) would be completely impossible given the large number of firms we're talking about.

I won't bore you with how it was actually done, but the bottom line is that it was chaotic and intransparent, especially in former Soviet republics like Russia. Insiders of companies, such as managers, often had priority in bidding for the companies - either formally, or informally through their networks. Oftentimes, they could purchase shares of the company for cheap and would make great money out of their assets. This is where the Russian oligarchs come from.

Of course oligarchs emerge from huge corporations in for example the gas and oil sectors. But the same applied to smaller firms such as ELORG. Now, I don't know anything about ELORG itself except for the documentary, but I can imagine Belikov having purchased shares of ELORG as a director, then also becoming the owner. As you can see form the docu, he also made alot of money from what once used to be state property (i.e. the Tetris license). The funny thing is that it Pajitov wasn't even an ELORG employee: ELORG just happened to be the state publisher and for that reason Belikov (rightfully, one must say) appropriated the rights of Tetris from Pajitov and the science lab to ELORG. Little did he know that five years later he would become personal owner and profiteer of that move.

To address your question directly: The Russian government didn't keep the Tetris licenses because it was part of the general transition to a market economy to dispose of state-owned firms and transfer them to private hands.

TL;DR: When the USSR/Russia turned into a market economy, firms like ELORG were privatized. Belikov became the new owner, probably by using his inside position (legally or illicitly), and made alot of money from Tetris. Belikov is a mini-oligarch.