I have a dog (black Lab) who has a phobia about climbing steps, compelling her to hesitate and then leap up over two steps to the front porch or back stoop, rather than just walking up them.
When overstimulated or frustrated, she will run madly in tight little circles.
She also likes to relax lying on her back with one foreleg stiffly extended in the fashion of a Nazi salute.
She is nuts.
That reminded me of my cat Ichabod who went nuts for frozen peas. And (less unusual, I think) I had a cat, Albert, who would go through walls for popcorn. We used to string popcorn for Christmas and let him have it after the holiday by stringing it around the room. He loved that so much.
Our now-departed eccentric Maine Coon, Midnight, used to like Corn on the Cob. If you were husking corn, she’d come over and demand her “cut”. She wanted several fresh kernels, right off the cob, placed on a bed of husks.
She also loved Vaseline, and would stick her head in a jar and lick it up. I gather this isn’t unique to her, though.
She also “spoke”. Hew cries sounded very much like words, and she seemed to use them that way. Pepper Mill was on the phone with the British Museum*, and the woman on the other end told her that Pepper’s child was asking to be let out. But it wasn’t MilliCal – it was Midnight, saying “Me—out! Now!” (which is how British Museum Lady interpreted it). All food was “Ham!”, which she enunciated quite clearly. Pepper Mill says that one time, when she opened the refrigerator, Midnight put her paw on the Thanksgiving Turkey that was defrosting there (in its plastic wrapper still), looked up at her, and said “Ham!”
Another cat that couldn’t care less about empty boxes.
And one of my previous cats weighed 27 lbs.
Zelda is the only cat who enjoys disassembling plumbing. Any small part that can be removed will be, and hidden so that we can’t find it. This includes the thing in the tap that covers the screw (which could not have been easy to remove) and the plastic bits that hide the toiled bolts and the ones that hold the seat on.
On the plus side, through her efforts the plumbers were able to see that our toilet wasn’t seated properly, just stuck on with sealant. She also periodically removes the hair clogs from the bathtub drain. (I trained her to do this by throwing a piece of ice into the bathtub to chase. When it eventually went down the drain, she scrabbled after it, and now checks every once in a while.)
I also made the mistake one day of throwing a kibble for her to chase. Now her bowl is not only food, but cat toys, and she’ll frequently fish a kibble out of her bowl and go chasing it around the kitchen. Her sister, whose main game is fetch like a proper pet, is baffled by the whole thing.
Our other cat Hestia expects a bowl of water on my nightstand, and will come out to wherever I am and cry if it’s not there.
I’m afraid that I got her started – I keep a glass of ice water on my nightstand to drink while I read in bed. She rapidly discovered this and started drinking from it, after which I wouldn’t want it any more.
I took to covering it with a card, but she simply realized she could just knock the card off and drink the water. At least the card acted like a warning system to me that she’d been drinking.
I soon realized that I could stop her drinking my water if I gave her another container to drink out of. This worked – she stopped knocking the card off mine. I eventually got a dedicated bowl for her.
The problem was that now she expects it to be there, and if it goes dry, or even too low, she’ll come and let me know about it. Whether she asks or not, if she’s in the room when I bring in a fresh bowl, she gets excited.
Of course, Hermes found out about it, and now he drinks from it, too. Hestia hates this – she regards the bowl as HER property, and she’ll chase him away if she catches him at it (despite his outweighing her significantly).
I had a dog that liked deodorant. When I’d put some on in the morning and then sit down on the bed to put on my socks, he’d try to lick my armpits.
I learned to put a shirt on before the socks.
I have a little Havanese dog who considers it part of her job to come and comfort me when I sneeze. She crawls on to my lap, turns her head sideways, and lays it on my chest. She does this at other times when she just wants to cuddle, but always when I sneeze. It is the most endearing thing I have ever seen.
I had completely forgotten until I read your post, but I had a foster dog for a while that did the same thing. I had to shut him out of the bedroom/bathroom when I got dressed or he’d practically tackle me to get to my armpits. I changed brands several times, but he seemed to like them all equally.
We had a Pug that went Bat shit crazy for tangerines. He could smell a tangerine being peeled from yards away and immediately want his share.
I also had a bulldog that would not eat alone. I would feed her in the kitchen and if I left the house, she would not eat anything. But as I get home, she would eat. And if I was not in the kitchen, she would grab a mouthful and bring it into whatever room I was in.
And she would eat every morsel…unless I scolded her for bringing the food into the living room. then she lose all the food in her mouth and I would be left to clean it up.
My sister’s cat sometimes acts like a dog, she’ll want you to be rough with her when you play with her. The dog on the other hand sometimes acts like a cat, he will put his head under your hand so you have to pet him. I love when this huge German shepherd grooms the tiny cat. The cat loves it.
My cat hates canned tuna and salmon.
My Morgan is like the OP’s dog with regards to the car. We go in the car almost every single day - if not to grammy’s house then to the park or the doggy beach. But every time we go out to the garage she hides behind the trash cans and I have to crouch next to the car door and talk sweetly to her until she comes near the door, where I then have to take her collar and gently nudge her to jump on in.
Her brother is the one leaving paw prints all over the car doors trying to get in. But Morgan won’t have it.
When it’s time to leave wherever we’re at, she is on a leash and I have to coax her to jump up while she feigns intense interest in sniffing the ground outside the car door before she reluctantly gets in.
I have a Labrador Retriever who refuses to retrieve a damn thing. For something supposedly hard wired into that breed of dog, she is an anomaly in that she has less than zero interest in fetching a damn thing.
My other dog, a Border Collie/Chow mix on the other hand will fetch tennis balls for hours and hours. 
My friend calls his black lab a “Triever”. She will run out after whatever he throws with interest, sniff it for a while, then walk back to him without it.
I had a rabbit, Bandit, who would eat the calluses off my fingers, and steal whole apples from my hand to nom under the dining table.
Our previous cat, Mrs. Tabitha Twitchit, was the most intelligent cat I’ve ever known - she was constantly doing things that betrayed an uncanny level of smart (for a cat).
For example: she would play fetch with a small toy ball, much like a dog. When humans weren’t around, she would bat the ball around on her own. She loved that toy ball beyond any other plaything. That wasn’t the clever part, though …
We noticed that playing with the ball had a drawback: inevitably, it would roll under furniture, where she couldn’t reach it. Then it would be lost until a human retrieved it.
Apparently, she noticed this too. And was determined to find a solution to always losing her favorite toy.
One day, we found the ball tucked into the hem of a translucent curtain we had hanging from a big window. The hem brushed the ground, and had a small slit delicately opened into the top, by unpicking the stitches. The toy had been placed into the slit.
I took the ball out and forgot about it.
Next day, it was back in the hem.
I took it out, and next day - it was back in.
Okay, I thought, she’s up to something … so I spied on her. I saw her put the ball back into the hem, holding the ball in her mouth and pulling the slit open with a paw. Then, she batted it around. She could still see at bat at the ball, because it was in the hem, which was gauzy and translucent - but the ball could not roll away under the furniture. She was playing ‘tetherball’ with it!
I thought this was very clever - she’d solved a problem (ball rolling away when played with) by deliberately unpicking the stitching on the hem, and placing the ball inside. This was no accident, as she repeated placing it several times.
Documented evidence that can be used against her… did she (or her lawyer) consent to being filmed? 
My former hound, Bucket, liked cows. If we’d pass a field while riding in the car, she’d intently scan out the windows to see if there was any livestock about. Once a cow (or horse, or sheep) was detected, she’d bark at it.
If she was looking out of the left window and there were cows on the right, we’d say, “Cows, Bucket” and she’d look around to locate and bark at them. If she happened to be snoozing, that phrase would wake her from her nap and she’d sit up and bark… even before the cows were positively identified.
My new mutt, Duncan, is not like this. I’ve even driven slowly along fields full of cows… he just doesn’t care.
My parents dog didn’t like “stick games” on TV. Golf, tennis or baseball would have him barking at the screen.
Our pug has a seriously Pavlovian response to the TV being turned off in the evening. Even if she’s asleep on the couch, she’ll leap up and run to her crate, waiting for her bedtime snack before curling up in her puppy cave. It may not be unique, but it’s adorable all the same!
I had a Gordon Setter named Fancy who loved Jeopardy. No other TV show attracted her at all, but no matter where she was in the house, when the Jeopardy theme came on, she would run out and jump on the sofa and stare at it. She would start to nod off and jerk herself awake, just like an old woman.