Just like you, your cat can suffer from things like anxiety. Your cat can't tell you that they are very anxious about something, so you have to know how to see what they are telling you in the way that they are telling you. If your cat is doing something like pulling out their hair, overgrooming to the point where they have bald spots or sores, and they are peeing and pooping in places other than their litter box, they may be having issues with anxiety. You can take your cat to your vet to diagnose them and to get some ideas as to how to treat their anxiety.
Again, like humans, your cat can take medication to help with their anxiety. That medication can include things like Prozac. It is pet safe, which means that it will be fine for your cat to take. You will have to give them a pill every day so that they will have the appropriate med levels in their system to make sure that they are having their anxiety controlled. While your vet is giving your cat a prescription for anti-anxiety medications, they may also give your cat a full physical exam to make sure that there isn't anything else that might be causing a problem, like allergy to a food or fleas. Some of the symptoms of anxiety can be the same as food allergies or other skin issues.
Another treatment you can try, either alone or in concert with other treatments is calming sprays and collars. These products have various things like pheromones and calming compounds in them to help your cats be calmer and less anxious. The pheromones would be the same kind of pheromones that a mama cat would be secreting for her kittens, and the calming compounds generally come from plants like chamomile and lavender, both of which have been used to help calm people for centuries. The collars are something that your cat can wear daily, and will need changed about once a month. The sprays you can spray around your house as needed. There are also misters that you can use, that you plug into the wall that will just automatically release the same things out to your house so that your cat can benefit from them.
If you are worried about your cat, you should take them in to see your vet so that they can be checked out. If they have anxiety, there are things that you can do to help them deal with it. Contact an animal clinic, like South Seattle Veterinary Hospital, for more help.