I think the term comes from how these regimes like to show power with tanks.
It was an insult made up by a group of British trots against other European communist groups for their support of the USSR intervention in an attempted color revolution (involving a bunch of actual og nazi-affiliates/Nazis) in Hungary in the '50s. The intervention involved driving tanks into Hungary, thus "tankie".
Of course later openings of CIA archives confirmed that the revolution was indeed supported by the CIA. The nazi thing was evident at the time, you can just look up the groups involved and the participants.