Are there any other animals that become attached to a non-living object?
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.