There is a lot of misunderstandings about service animals and support animals.
Service animals are those covered under federal law–these animals are trained to perform a task for a disabled person.
Support animals are those covered under California state law and are completely different. They may be an animal that helps an individual cope with depression, or may perform a task. However, no special training is implied or necessary with a support animal.
The federal law addresses public places, such as a mall, movie theater, lobby of a building, etc. The state law addresses private places such as places of employment or housing.
Regardless of whether it is a service animal or support animal, in most situations you must allow tenants to have them as a reasonable accommodation.
There is a lot of confusion about what constitutes a service animal or a support animal. In the past service animals were easily identifiable because they were often with person who was blind or in a wheelchair. Now, you often cannot tell if it is a service/support animal because so many disabilities allow service/support animals that they are so much more common.
If you have any questions about this, or how handle the situation of a tenant with a service/support animal, please be sure to seek out an experienced property manager or an attorney. Failure to follow the laws exactly may result in a lot of problems for you.