I think people with the Pal Mickey and the newest ones in the croc exchange do it for fun. It's very funny to read those stories.
I would hope no one would do that for real though. However, years ago and friend and I went to a Teddy Bear show. People had their bears all dressed up and had some in strollers and/or baby carriers strapped to their bodies!
Okay, I have a few things to add.
I take part in an exchange similar to the croc one, except it's a monkey. We didn't take "him"

with us every day, but we did take him to
Castaway Cay and to the parks one afternoon. It was so much fun to do. He tried on our pins, our flippers and scuba gear, he was found in the drawers and shower of various resorts, drinking a Conch Cooler, etc. It was a really silly and fun thing to do. I can see why people enjoy this, esp. if you are a solo or a couple. It is like wearing Mickey Mouse ears or a tiara to me, just something fun to do to lighten up the day and enjoy being away from the normal, serious, work world.
I don't think it is a sign of any mental problems, but I can see how someone might become dependent on it. Now, many people are dependent on things - food, cigarettes, alcohol, work, cleaning, drugs, shopping, plastic surgery, etc. Everyone has things that they go overboard on. It's only a mental illness if it affects your developing friendships or having a happy, safe and productive life. If relying on stuffed animals interferes with that (as a previous poster mentioned about a suicidal friend) then we have to hope that they will all have a friend or family member as great as that poster to help them out. We can't tell from looking.
Also, I am a former CM and while working, I noticed some unusual things. There was one repeat guest who brought a stuffed dog with her to the parks. Not a plush dog, a real deceased dog that had been taxidermied. It smelled so terrible, even from a distance. She held it next to her in her tote bag, with its head sticking out as if it were alive. She stayed at the show where I worked the whole day and watched every single performance from opening to closing. It was the only time I ever refused to do something at work. We switched positions and I was supposed to stand very close to her and I just couldn't. I said, put me anywhere else, but this woman's dog is the same breed and color as my former childhood pet and I cannot deal with that. I felt bad for her but it hit too close to home. I don't know if it interfered with her life negatively, but it certainly did mine!
So, plush stuffed animals, no problem. Real stuffed animals? I can't handle that!