What is it about Soviet born players that makes them winners in the brutal 128-player knockout format of the World Cup? Teimour Radjabov has just won the World Cup.

The first two FIDE World Cups, in 2000 and 2002 had 4 groups of 6 players with the 2 group winners going into an 8-player knockout. They were both won by Anand and were stand-alone tournaments.

In 2005 the tournament morphed into its current form, a 128 player knockout with a place in the Candidates as the prize (apart from large bundles of cash) played every 2 years. 

Knockout is a brutal format which favours the toughest players over the very best. Even though the Soviet Union ceased to exist in 1991 every single winner of the current format World Cup was born in the former USSR. Here is the list:

2005 - Levon Aronian (born 1982 in Yerevan, USSR)
2007 - Gata Kamsky (born 1974 in Novokuznetsk, USSR)
2009 - Boris Gelfand (born 1968 in Minsk, USSR)
2011 - Peter Svidler (born 1976 in Leningrad, USSR)
2013 - Vladimir Kramnik (born 1975 in Tuapse, USSR)
2015 - Sergey Karjakin (born in 1990 in Simferopol, USSR)
2017 - Levon Aronian (born 1982 in Yerevan, USSR)
2019 - Teimour Radjabov (born 1987 in Baku, USSR)   

The very best non-USSR players have entered multiple times and failed. Players like Carlsen, Caruana, Ding, Anand have entered but failed to make it all the way though the tough competition.

What is it about Soviet born players that makes them winners in this format?
Most top participants were ex-USSR. Almost all of the World Cups were held in ex-USSR countries. And the rest is coincedence.

Especially in earlier years, most of the top participants were born in the USSR.

E.g. in 2005, the top-25 by rating of the participants was Ivanchuk, Bacrot, Aronian, Grischuk, Gelfand, Shirov, Akopian, Radjabov, Ponomariov, Tiviakov, Sokolov, Dreev, Kamsky, Bologan, Lautier, Bruzon, Bareev, Mamedyarov, Vallejo Pons, Smirin, Harikrishna, Malakhov, Sakaev, Volokitin, Moiseenko.

Only 6 in 25 (Bacrot, Sokolov, Lautier, Bruzon, Vallejo Pons, Harikrishna) were not born in the USSR. And only 1 in the top-10! And that tournament had Kasparov, Kramnik, Anand, Topalov, Svidler and Morozevich (4 out of 6 USSR) missing because of world championship situations.

In later years the number of non-USSR players grew slowly, but in 2011 the first non-USSR player on the list was #12, Zoltan Almasi.

2015 was the first year with only 4 USSR players in the top-10, but 11 in the top-20 (#11 Karjakin won).

It's only in the last few years that it's a bit against the odds that a USSR player won it.

There are of course top non-USSR players: Carlsen, Anand and Topalov were consistently in the top most of this period. But as the World Cup was mostly a qualification event for the Candidates and they qualified otherwise (or were World Champion), they played in very few World Cups.

Apart from 2013, all World Cups were held in USSR countries

Of which three times in Khanty-Mansiysk, Siberia, sometimes in november/december. I think Russian speakers have a psychological advantage in Siberia, they feel more at home because they can at least understand the people.

Coincedence

Taking the facts above, this is easily within the realm of coincedence. I wouldn't bet on a ex-USSR winner next time purely based on the results so far.