Specifically we would like our Mobile Hybrid App to be able to receive Push Notifications from Salesforce MarketingCloud MobilePush, but to also have the ability to send Push Notifications, using the same Firebase account and credentials, programmatically from a non Salesforce piece of software.
This should be possible, although there has been no testing around this use-case with the Cordova plugin.
The native platform SDKs for the Marketing Cloud support multiple push providers so you should be able to accomplish a setup where it will work in Cordova. It would be up to you to implement the necessary native code for each platform.
Using the SDK's documentation on multiple push providers should get you going in the right direction.