First, a 'service dog" is still a dog and can cause damage. Usually this is because of negligence from it's owner: failure to properly groom (cut nails, etc), failure to properly walk (for defecation/urination), etc.
Second, "service" dogs are trained from a very young age. They are screened for temperament and ability, then rigorously tested and certified. A pet does not "become" a service dog.
Third, a certified service dog only allows the tenant to have one, it does not release him from the liability/obligation of pet deposits, damages, etc.
Fourth, a "support" animal is a scam propagated by the liberal mental health industry to allow people to skirt the laws. Now, some people are mentally imbalanced and need a fluffy thing to calm them, but 90% are just using it as an excuse to skirt the law.