Do communications between an EMS provider and patient have the same legal status as communications between a doctor and a patient?

It depends on whether the Paramedic is working for a private or government agency.

Yes. Privileged communications are always considered an extension of the doctor's professional license under which the paramedic practices.

It depends on the state law where the Paramedic is practicing emergency care.

No$.$ Communication during an emergency is never protected by any type of privilege.