We do. Of course, we can never know which one will end up with something, though. Bird, a feral cat that we took into the house and had until we lost her a year ago at the age of 23, was a toy-player. Throughout her life, that cat knew how to play with a toy, and she claimed them all. When she was still an outside kitty, during the time that I was feeding her but hadn't taken her in yet, I watched her batting a large inflated ball around that she had found somewhere. It was between the size of a softball and a beachball. When she was done playing with it, she pushed at that thing until she had it wedged under the vacant house that she was living in. She'd never be able to get it out again, but neither would anyone else. It was her's.
When we bought toys for our cat, Bird would claim every one of them. Periodically, she would round them all up and store them in one place, whether under a couch, a corner of a room, or even a box. It was hilarious. She would put away her toys, and everyone else's.
I blamed my nephew, who we were raising, for stealing a pen from my desk. Later, I watched Bird jump up on my desk, pick up another pen, and jump down with it, claiming that as her's as well.
Her first Christmas in the house with a Christmas tree, she knocked it down in short fashion. After we secured it better, she'd climb it and pick ornaments off the tree. The tree was in the upstairs living room and my wife and I were downstairs when we saw Bird coming down the steps with a wrapped present in her mouth. She was something else, and I really miss my Birdie.