Comfort. Safety. Security. For children, these are all synonymous with love. Children have their binkies. Their blankies. Name a nonsensical thing, and I'm sure there's a lovie out there with the name. For our kids, comfort objects are even more important to their development. They need something they feel is secure. They need something that soothes them. Period.
So, how do you support your kid's need for their lovie of choice? What's socially acceptable? First, you present some to them. And keep presenting things until something clicks. They need that security that they equate to love. And then you respect it. Socially acceptable behavior is something that happens over time, over the course of our children's various therapies, and exposures to other families and kids. In my opinion? It's important for them to have the security... and then eventually the security to let it go.
At least you respect it until it doesn't look like the object it is, and you become determined to get that sucker into the washing machine, pronto. And then come your maneuvers in the middle of the night worthy of Mission Impossible to get it clean before the kid wakes up. But, I digress.
So, what happens when you go out into public? The younger the kids are, the less likely you'll get the look. But what about when their older? I don't have any concrete recommendations, to be honest. I've learned to have a hard skin when it comes to that sort of thing. As far as I'm concerned, the people who stare/judge can go live their own life - sans lovie. However, you do need to gauge your child. They do need to make progress towards behaviors that will serve them later in life. Do they still need it? Do they need something else? Are they ready to move on during the day, but still need it at night in bed? Let them lead you. Experiment. It's a good lesson for them in gauging what it is they're actually feeling. The bottom line is - the thing gives them comfort and security. Like most things in parenting... follow your child's cues. There aren't any rules here. Maybe you move from one kind to another - blanket to fidget for example. Perhaps the comfort item becomes an at home thing, and you can substitute a different, more socially acceptable item out of the house.