In Russia, Russian Orthodox Christianity is the religion most associated with red clothing in church settings. In that tradition, red vestments are commonly worn for Easter and other solemn services because red symbolizes Christ’s blood, martyrdom, and resurrection.

What the red means

Red is not usually a sign of a separate religion; it is a liturgical color within Orthodox Christianity. In Russian Orthodox practice, different colors are used for different feast days, and red is one of them.

Other possibilities

In broader Russian folk tradition, red also appears in traditional clothing and ritual dress, but that is cultural rather than a religion by itself. So if you mean a religion practiced in Russia that uses red clothing, the best answer is Russian Orthodox Christianity.

TL;DR

Red clothing in Russia most often points to Russian Orthodox church vestments, especially for Easter and other major feast days.