Nobody “owns” it. It was hidden in a stone wall at Antioch for hundreds of years until an earthquake exposed it. Franks took it back to Europe with them and it bounced around Europe in possession of various families for a long time.

As for Russia, they (and I) venerate an Icon of the Mandylion which is the face from the shroud.

Ever heard of the Suderium of Oviedo? It’s the face cloth mentioned in the Gospel accounts, and the stains on it exactly match the image on the shroud.

It’s real, and it’s spectacular. (To paraphrase Seinfeld).