Cats dislike cages for many different reasons. Some cats probably associate the cage with unpleasant things such as going to the vet for checkups or procedures. It is also possible that the cage smells like the vet, or some other odour that brings up unpleasant memories for your cat.
Here’s a video example of a cat that really really does not like to be in a cage. This video currently has over 60 million views on youtube. Note: The video has a disclaimer stating that the cat is not abused — it simply does not like the cage.
My oldest cat likes cages and likes cars, but she dislikes being in a cage in the car (go figure!). When the cage is at home, she will let herself into the cage and sometimes shut the door behind her so the other cats cannot join her. She also likes cars. However, if she is in the car, in a cage, she will usually take a poop and then roll in it. Very gross! To make matters worse, she is one of those cats that really dislikes getting wet, so she usually doesn’t enjoy the cat bath that follows.