In short: There is currently no way to change the firestore database to write to from default.
See question on stackoverflow: https://stackoverflow.com/questions/78533417/how-to-write-to-a-firestore-db-other-than-default-in-a-python-cloud-function

As far as I understand there is currently no option to pass the databaseId to firestore.client or to the options when initializing the app (initialize_app(cred)). You can pass the databaseURL to the options in initialze_app, however the Url cannot be looked up in the firebase console anymore.

In my opinion, it would make sense to passt database as an argument to firestore.client() much like you would when initializing the Client class in google.cloud.firestore directly.

db = firestore.client(app, database="my-db")

Alternatively the databaseId could be passed as an option when initializing the app.

app = initialize_app(cred, options={"databaseId": "my-db")

I'd be willing to implement this feature.