By Farah Stockman

A 26-year-old Starbucks barista in the suburbs of Tampa known as Vayne Myers has suffered from anxiety ever since he was a child. A co-worker suggested he try an emotional support animal.

So Mr. Myers bought a duck and named it Primadonna. The snow-white bird has worked wonders for his state of mind.

“Whenever I felt like I didn’t matter in the world,” he said, Primadonna would waddle over and remind him that “something does love you.”

But Mr. Myers’s landlord objected, and demanded proof that Primadonna was a medical necessity and not simply a pet. Mr. Myers provided a letter from a therapist in California who spoke to him over a video chat, and then another note from a counselor who met in person with him (and the duck). But neither document satisfied the landlord, who threatened eviction.

Mr. Myers hired a lawyer and filed a complaint of housing discrimination with the Department of Housing and Urban Development using his legal name, Jesse Calfas. His filing was one of more than a thousand similar complaints the agency has received nationwide so far this year.

The number of people claiming they have a right to live with animals for their mental health — as well as to take them onto planes and into restaurants and stores — has been growing rapidly.

In 2011, the National Service Animal Registry, a for-profit company that sells official-looking vests and certificates for owners, had 2,400 service and emotional support animals in its registry. Now the number is nearly 200,000.