There are three things you can ask under ADA (which does not allow ESA, and appears to be what you are talking about, FHA and ACAA have different rules):

0. Is this a dog or miniature horse? (only those animals are considered as potential service animals under ADA)

1. Is this a service dog within the definition of the ADA, i.e. has it been trained to perform a specific tasks to mitigate a disability? (cannot ask what the disability is)

2. What specific tasks is it trained to perform? (ex. seizure response, seeing eye, etc.)

However, there is no way for you to "check" if this is true. Service animals need not be trained by any approved company or carry any identification, and you cannot ask about this. The only way out is if someone admits to it not meeting the legal definition or gets caught in a lie while answering these questions.

If, after all that, the animal cannot be reasonably accommodated as it is not being housebroken, being vicious or harmful, damaging property, or making an unreasonable and excessive amount of noise, it can still get shown the door. Likewise, if allowing an animal is inappropriate under any circumstance for safety or regulatory reasons (ex. cleanroom, food service, nuclear exposure, hearing damage potential, etc), you need not allow it. The establishment owner is never responsible for feeding or looking after the dog or procuring one for the person with disabilities. You can expect that service animals be kept on lease unless brief exceptions are critical to job function.