My cat gained weight but eats normally. Is it thyroid or diabetes?

Cats get hyperthyroidism and this actually causes them to get too skinny. Cats who are over weight at at a much greater risk for diabetes but if they get the disease, they tend to lose weight and drink excessively. To answer your questions though, both conditions require blood work for diagnosis. Most cats get heavy because they eat too much and their metabolism slows down.
Feeding multiple cats is difficult when one needs to lose weight and requires commitment on the humans part. You cannot leave food down all the time. You must separate the cats at feeding time. You must switch the heavy cat to a light cat food and control the portions. Cats do best if they are feed 4-6 times a day. For my own fat cat, I got an automatic feeder that stays shut in a room. It portion controls and is timed. All I do is open the door and shut it behind him. It’s pretty easy and I like that.
I hope this is helpful and please feel free to post any additional questions.