As far as I know, no major airlines have any special treatment for "emotional support animals." Most U.S. airlines allow pets on domestic flights to fly if they stay inside carriers within approved size limits. Emotional support animals and therapy animals fly as pets regardless of any certifications. So I'm pretty sure there's no service that makes it easy to fly with your Great Dane as an emotional support animal. You might be thinking of other animal-related exceptions, like having a pet in your apartment where the lease normally doesn't allow pets.

Service dogs on commercial flights are a separate USDOT category. The dog needs to be trained for a specific task for a disabled passenger, and the passenger must provide an attestation form. Airlines must allow service dogs, but they can still deny transport if the dog poses a safety risk or causes significant disruption before or after boarding. I'm not sure how enforcement works in practice, but I certainly wouldn't try to fly with a dog using a false attestation.