My grandson is six and this is the first Christmas I am NOT giving him a stuffed animal as a gift. He has about 30 stuffed animals and they now hang in a web-type bag above his bed. Up until a year ago he slept with one or more every night, and often carted one to my house when he visited. Here's what happened...
He's in first grade. He hangs out with first grade boys, and older, on the playground. None of them are into stuffed animals - that's 'baby stuff.' He also says the cartoon shows he once enjoyed are 'for babies.' When I ask where he got that idea, he said 'all the guys' say that.
In other words, he's growing up. In some ways it's sort of sad, however he doesn't make fun of his two year old brother's interest in stuffed animals, and sometimes I see him playing with the brother - with the toys - making the lion 'eat' the little brother or tickling him with a stuffed rabbit, etc.
So if a six year old boy still enjoys stuffed animals, I'd say he plays with them only at home, with a younger sibling, or when alone.
And definitely not when he has the 'guys' over to play.