They were receiving major persecution from the Tsar.

The worst thing the Tsar ever did to the Jews was to try to force them to do manual labor. He gave them land and plows, and for that they wiped out his entire family.

Can anyone provide some info on the connection? Is it overplayed? Is it true?

Communist movements were extremely Jewish, especially before Stalin consolidated power (even though he cracked down on some Jews, they still had significant influence - his right-hand man was a Jew named Kaganovich). Jews made up 60% of the leaders of the German Revolution. Many other Communist leaders were Jews, including Lenin, Trotsky, and Bela Kun. Jewish Wall Street billionaires (Jacob Schiff, the Warburg Brothers) were also involved in financing these movements. "In fact, 85 to 90% of the foreigners helping the Chinese at the time of the Communist takeover were Jewish."

Why does it persist even up to its current form of "Cultural Marxism".

This is a means of capturing Judeo-skepticism and converting it into support for mainstream conservatism. The far-left cultural shifts do follow the ideas of the Frankfurt School Jew Herbert Marcuse, but these cultural changes mostly haven't taken place in Communist or former Communist countries. This term obscures the corrosive effect of liberal democracy and free-market capitalism.