Service animals can and do support people with emotional/psychological health concerns, including PTSD and panic attacks. They are still service animals and not ESAs. A service animal is trained to perform specific task(s) that help a person with a disability. When on duty, they are working. ESAs are different not because they do not help, but because they do not require any type of task performance: an ESA's presence is their benefit. A friend has a cat who is an ESA, and the cat's companionship and existence (e.g., the things the friend has to do in order to take care of a cat) are the benefit. Another friend has an SD that is trained to interrupt their compulsive behavior cycles and their self-harm rituals. Both deal with psychological disabilities but that does not make them the same.

historically ESAs were legally required to be allowed on planes--not under the guise of "service animals" but specifically as ESAs.

there is no required certification for other service animals (which in the US are exclusively dogs and miniature horses). while an effective and appropriate SD team will have a lot of training and so often has some certs, there is no specific requirement, legal registry, etc.