I've seen plenty of people do this! I've also seen people put little stuffed animals in the cage with them too.
Mine doesn't have it in his cage but I will always pull out a blanket when I take him out for him to sleep on and snuggle in. He loves snuggling in my fuzzy shirts too!
I thought about a washcloth.. I've just noticed him scratching in his sleep area before settling in for the night, thought a soft spot would be nice for him.. I think I'll try the washcloth tomorrow..