FYI: FAA has different rules regarding service dog vs. support dog. Due to this, many people use this difference to bring their dog in the cabin with them. This is a "popular" reason for claiming "support dog" status.
Most stores, restaurants, etc. are required to follow the ADA rules and can therefore deny "support dogs" but not "service dogs" (and there are stringent requirements regarding what designates a service dog). Unfortunately, many stores, restaurants, etc. employees who have to deal with the issue aren't aware of the law, don't have the support of their management, etc. to hold their ground when someone brings their dog inside.