Run a web socket on Cloud Functions for Firebase?

Theoretically you could use two different layers: one to manage the websocket connections and another layer to handle the data processing.

The websocket layer will not be Cloud Functions, but a Docker container running Push Pin in Cloud Run and that’ll route HTTP calls to your Cloud Functions to do the actual data processing.

This is not going to be a good fit for Cloud Functions. Websockets rely on long-lived connections to the same server over time; Cloud Functions are ephemeral compute instances that are spun down when there's no traffic. There's no way to force or guarantee that a Cloud Function will keep running or hold a connection open indefinitely.

I would encourage you to investigate using the Firebase Realtime Database as a conduit here instead of trying to add realtime to Cloud Functions.