KAZAN, Russia—A statue here in the capital of Russia’s Tatarstan republic was meant to symbolize the intimate links between this Turkic-speaking region and Turkey.

Born nearby, Tatar politician Sadri Maksudi was a leader of Russia’s Muslims a century ago. Then, forced into exile by the Communist revolution, he became one of the closest advisers to Mustafa Kemal Atatürk, helping him create the modern Turkish state.

A...