I have had several people over the years ask me about writing them a letter for an ESA. It's an amazing concept and there is research out there that can support the positive benefits that pets can have on a person's health; some of those benefits can be decreasing stress, lowering blood pressure, and increasing your mood. You can read an example of that research here. There is also research on service animals and they can be trained to help support those with disabilities. However, there is really no research or evidence for an ESA specifically, and how they can support those people with emotional stability. While having an animal companion may be beneficial, let's look at some of the criteria and limitations for having one.
The only requirement for an animal to become an ESA is a letter from a trained professional stating that your pet is an ESA. This is a gray area because most professionals don't have the ability to determine if your animal is proper ESA material. They can attest to how your mental health is; and with the research on pets, it could make sense to give this as a suggestion for you to look into. Many therapists won't write letters as a result. Some will, though they may have specific requirements for doing so.
A service animal, on the other hand, is trained to perform a specific task that can help someone with a disability. This can be physical or mental/emotional. For example, if someone is blind, the service animal is trained to be able to lead them without bumping into things. For someone who has PTSD, a service animal can be trained to increase feelings of safety by performing room checks, or even supporting someone if they are having a panic attack. They are also trained to be well behaved, so they are allowed everywhere you go. Because an ESA is not trained, it is difficult for many public places to allow them into their business, with the exception of housing or airlines. Keep in mind, with both service animals and ESAs, there are rules, especially if the animal is causing harm. Additionally, if the animal is misbehaving, it can create more anxiety or increase emotional struggle in the owner, which is the opposite reason one wants to have an ESA. While there is potential for a service animal to misbhave as well, it is less common due to the extensive training that they are required to have.
Many people opt to have an ESA instead of a service animal because they either do not know that there are service animals available for mental health and believe they are only for physical disabilities, or they financially cannot afford to train a service animal. It can also take a long time for your animal to be a service animal due to training, which may not be ideal for some people. If they are in training, they aren't always allowed in places that a service animal would technically be allowed.
So overall, many people agree that animals/pets can have a positive impact on one's mood/health. If you are thinking you may benefit from having an animal companion, do the research and decide which one is best for you.