Yes some cats do suffer from separation anxiety when they've become close to their owners.
If a cat is suffering from separation anxiety the cat may meow excessively or even urinate outside the litter box as well as not eating or eating too fast or self grooming excessively as well.
Cats will usually get over the separation anxiety overtime but you can get the cat another cat of the same gender to help keep the cat from becoming lonely and anxious while you're away from the home.
Or if you do get a cat of a different gender then be sure that both cats are fixed so they can't reproduce.