YAROSLAVL, Russia—Yuri Novikov feels the Bern.

The 78-year-old Russian surgeon remembers the day in 1988 when an unruly-haired Burlington, Vt., mayor named Bernie Sanders and his new wife, Jane, then director of the city’s youth programs, showed up in this thousand-year-old city on the Volga.

Formally, the American couple had arrived in the...