Do any animals become psychologically attached to any non-living object that is not useful to the animal for any of its physical needs, like some children do to stuffed animals? Some children are so attached to their toys that they take them with them whenever possible, and are upset when the toy goes missing or is taken by another.
So, this is psychological attachment. From the question alone, I thought of physical attachment.
If you provide stuffed animals and human children, then I'd offer the television remote, a chair/sofa and the cellphone for human adults. In general, many adults are far too sedentary for their own health.
Some non-human primates have exhibited behavior similar to that which you describe with the stuffed animals-children example.
In terms of an inanimate object, you can add the Internet. Certain behaviors like over-indulgence in alcohol, sex, gambling, hand-washing, and the like, could, broadly speaking, fit within the question as posed.