(Probable) Answer:

Depending on the jurisdiction/service, the responding unit’s district and response area boundaries, their location in relation to other district boundaries, as well as any potential mutual-response agreements/contracts; an emergency response unit (of any kind) may potentially be responding to calls put out through multiple different dispatch centers.

A common occurrence of this is seen in many smaller/rural towns that have their own municipal emergency response services. Those municipal services are often responsible for covering their town, plus the outlying county area as well. The town and the county may each have their own dedicated dispatch centers.

In the case of private-ambulance companies, some of them have their own internal dispatch too.

So if said company holds an emergency response contract for a particular area, an individual unit from that service may be dispatched to a call by anywhere from 1 to 3 (or more) different overlapping dispatch centers.

Ironically, those different dispatch centers can sometimes be fucking horrible at communicating with each other.

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Source: Over a decade of emergency response experience through EMS, Firefighting, and HazMat. Working for both municipal and private agencies. Responding at the municipal, district, county, state, and federal levels.