It certainly can not mean that the tens of thousands of other clinics can not comply. I don't believe I have ever been in a clinic that was not governed by the state board of vet med.
While a non-accredited hospital may actually be in compliance, a huge number are not. There are mandatory minimum standards and then multiple suggested standards ranked by a point system - a minimum score in about twenty different categories is necessary to be accredited.
Every state indirectly regulates veterinary hospitals through licensing of the veterinarians. Many do not inspect individual hospitals (unless there are complaints to the vet board) or even have standards representing minimum standard of practice.
One thing that made veterinary news recently, Alabama is a state that requires inspection and will now accept AAHA accreditation/inspection in lieu of state inspection. I think California and Tennessee have inspection (don't know how often) and I'm sure there are more but it's not universal.
Board certification in vet medicine is a different ball game than human medicine. When I graduated from Vet School, the only Board Certifications were in dermatology, ophthalmology and surgery. Referral to specialists was just about never done. Now there are boards in just about every discipline and referral is common. Being board certified means special training and education in that specific discipline. There is no need to be associated with an AAHA hospital, but if you look most are.
Unless you have a specific problem, a good GP veterinarian (board certified or not) is a good place to start.