Rather than continuing to hijack this thread (or keeping it restricted to cats), I thought I’d start a thread so we could share strange things that our pets have enjoyed eating (be it human food or otherwise).
I had a lab/chow mix that liked chips and salsa, as well as cigarette butts, aluminum cans, and beer. My present cat really likes french fries for some strange reason.
My dog likes vegetarian cheese
One of my boxes loves dill pickles. The other is absolutely crazy for raw potatoes. If you’re standing at the counter, peeling them, next think you know, you’ve got this sad eyed doggie looking up at you. He LOVES them.
One of my old cats was crazy for KFC chicken. He didn’t like the skin, just the chicken. But, if it was chicken we’d made, no go.
My weimaraner went nuts over raw broccoli. We would give her the stems and she would devour them and beg for more. She got ahold of a jalapeno pepper once; she spit it out and barked at it.
When I was a teenager I had a persian cat that loved peas. She would eat them one at a time.
My dog LOVES carrots and asparagus.
My cats like cold ,canned asparagus, dried banana chips and cheddar and sour cream potato chips.
I used to have a cat that loved Peanut Butter Crunch cereal, spinach and she would tear your arm off for an oatmeal cream pie.
My late lamented springer spaniel couldn’t get enough watermelon. He would go nuts as soon as he knew there was one in the house. He ate all the rinds when we were done and was in heaven if we let him have a whole piece.
My dog likes fruit. Apples, okay, oranges sometimes, grapes, pears, watermelon, but he hates banana.
He, however, LOoooooooooooooves papaya. One whiff of that, and you have a droooling dog in your lap.
No idea why he likes papaya so much. 
I’ve got a friend whose dog loves green peppers and cucumber. Strangest things.
My cats (well two of them, one won’t go near anything “people food”, and tries to cover it with litter or towels if we give him any) love anything white. Milk, ice cream, mayo, marshmallow, whatever. If it’s white, beware.
Randall, the youngest, is a fiend for Arby’s roast beef. Tank, the oldest, is a fiend for roast pork.
The dog is just a dog. But don’t say the word “milk” or she won’t leave you alone until she gets some.
Most of the cats I’ve ever had have shared my love of popcorn.
Just the noise of the popcorn pan gets Cuervo running into the kitchen to beg.
My dear departed Rocky, who was the best cat in the entire world, ever, loved olives. He was the most gentle creature, but would kill for an olive. If allowed to have one, he would hook his claw into it and chomp away.
Our dog loves frozen vegetables. And ice cubes.
My dog loves broccoli, too. We once had a vegetable platter on our coffee table when we had guests over, and we turned around and found her gobbling all of the broccoli there.
Our cat will kill for french fries from McDonald’s or Burger King. She only eats one or two, but it is heaven/drugs to her.
She could not care less about other french fries. Our faves are the ones from Wendy’s, and she smells them and goes bonkers for a french fry, but when we offer her one, she sniffs, paws at it, and leaves. So the smell triggers something, but upon closer inspection, she finds them Not Worthy. No clue as to why.
A friend of mine has a golden retriever who loves sauerkraut.
Some cats like melon. Leucous, who owned me for about fifteen years, loved any kind of melon except watermelon. Reuben, my co-owner at the time, would watch in puzzlement as Leucous devoured his canteloupe.
Cindy, the family dog when I was a kid, loved raw carrots. I would bite off pieces and throw them to her. She would catch them before they hit the floor and crunch away with bliss.
Maggie will eat anything that is people food. Corn chips, potato chips, crackers, toast crust, you name it. I mentioned before that she loves donuts. She will chew through a plastic bag to get to said donuts.
She’s tried to pull the lettuce and cheese out of my sandwhiches. She’ll paw at french fries to get them off of the plates. One time she chewed up a hot dog-only to ralph it right back up.
She thought that pepper was trying to get her…and she was right, it was! 
I had a cat that loved corn.
<hijack>
But cheese is made from cow (or goat or sheep) products. When it comes from an animal, how can it ever be “vegetarian”?
Or are you using one of the ‘relaxed’ definitions of “vegetarian”?
</hijack>
You know how people grow zucchini in their gardens and end up with dozens of giant monster vegetables, and everyone is like “what are we going to do with all this zucchini?” My cousin’s chocolate Lab loves them. I watched in amazement as this dog gobbled down a foot long zucchini in about a minute.
I used to have a cat who loved mint. Toothpaste, candy canes, starlight mints, anything. A wintergreen Lifesaver once melted onto the concrete on our front porch, and over a week or so he licked it clean.