No it doesn't mean they can't work with multiple landlords, it means the represent the landlord(s) interests, not the tenants.
If a tenant hires his own agent to find an apartment, that agent would (a) represent the tenant and (b) the tenant would pay for that service.
But I'm not sure what stops the landlords from just raising the rent to cover the broker fees. Unless there's enough competition from landlords who don't use brokers.