Why is there still a government-in-exile for the Belarusian People's Republic, a country that has not existed for a 100 years?

I'll end this answer at 1996, but the short answer is that the transition for most Soviet states to market economy was drastic, save for two: Belarus and the breakoff republic of Transnistria.

In places like Russia, Kazakhstan, etc. the elite simply became oligarchs and security agents, formed inner circles and put on a new face to an old elite. This is of course an oversimplification: numerous young, hustling entrepreneurs rose to become billionaires through a mix of shrewdness, being in the right time period, and corrupt deals. On the other hand, thousands of formerly powerful bureaucrats were relegated to jobs like taxi driver or store clerk.

That said, in Belarus and Transnistria, Communist elites stayed in power, and never drastically changed the system. State ownership prevailed, the military remained in Soviet style, and politics remained one-party under a barely renamed, barely repackaged Communist party. Newspapers called Belarus the "last dictatorship of Europe" and the system was still distinctly Soviet.

In Kazakhstan, President Nursultan tipped his hat to Kazakh nationalists singing Turko-Mongol folk songs and waving their banners. In Uzbekistan, Islam Karimov built hundreds of statues to Timur the Lame, honoring the nation's pre-Communist history. In Russia, Yeltsin and his group welcomed back descendants of the Tsar, and the Orthodox Church sought to glorify the monarchist past.

There was no such reconciliation in Belarus. It made no sense for the exiled government's supporters to come back, because the Belarusian government never invited them back or sought to erase the Soviet legacy.

I won't comment on developments since 1997, but at least in the 1991-96 period, one could say Socialism never ended in Belarus.