Mine has a chain. Why speculate? Lower the spare and check.
It's almost certainly the chain - it has to be long enough to lower the spare to the ground, so the chain ends up hanging down through the spare. Some times it makes enough noise to annoy me, so I stop and stuff the chain back up into the spare.My wife called me right now and it said something was hanging from the spare tire. I'm at work at the moment. It might be the chain like yours.
It's almost certainly the chain - it has to be long enough to lower the spare to the ground, so the chain ends up hanging down through the spare. Some times it makes enough noise to annoy me, so I stop and stuff the chain back up into the spare.