While there isn't a HIPAA for veterinary medicine, many individual practice acts have privacy stipulations related to medical records and veterinarians can get in big trouble for violations. I'm not sure how it works if staff is responsible for violations (probably varies by state) but a big deal nonetheless. The AVMA has a decent (if brief) review of some laws in different states
here.
On a similar note, something like this is a serious ethical violation and can have serious consequences for clients. I have personally worked with clients who were hiding from domestic abusers etc who could have easily been found and murdered if their/their pet's information were publicly shared like this. Even barring unique situations like that, given the harassment experienced here I wouldn't be surprised if this client had enough to file charges or pursue a civil suit depending on the location.