Answer:
It could be both.
In your community, the association office may be a facility in the same real estate category as the pool house or the open parking garage. As a facility it is identified in the reserve study, maintained with assessment dollars and owned by all owners. (Unless your governing documents define it otherwise.)
If the office is off-site, the office may be rented.
As a service, the association's business is conducted in the office space. The service collects assessments, pays bills, identifies and tracks maintenance issues, handles property management logistics and so forth.