Thanks @bgilmer. What I found is after initially adding the the openHAB Cloud Connector add-on through PaperUI, it will show the error Socket.IO error: not authorized in the karaf log.
This is apparently expected, as the secret and potentially also the uuid have just been generated. It attempts to connect to the default Base URL for the openHAB Cloud server at https://myopenhab.org/ but fails since we need to associate an account with the newly generated secret.
I proceeded to create an account on https://myopenhab.org/ with the secret and uuid found in these locations:
openhab/userdata/uuid (or via REST API rest/uuid)
openhab/userdata/openhabcloud/secret
After that we need to trigger the openHAB Cloud Connector add-on to try and reconnect to the cloud server. You can either restart openHAB manually, or in my case you can simply go to PaperUI > Configuration > Services > openHAB Cloud and click “Save” on the dialog without changing anything.
Once I did this the karaf console show a successful connection and I am able to remotely access my openHAB2 instance from the web or the mobile app.
[INFO ] [io.openhabcloud.internal.CloudClient] - Shutting down openHAB Cloud service connection
[INFO ] [io.openhabcloud.internal.CloudClient] - Disconnected from the openHAB Cloud service (UUID = 73a0a571-e48a-4ccf-af64-5342020b05a0, base URL = http://localhost:8080)
[INFO ] [io.openhabcloud.internal.CloudClient] - Connected to the openHAB Cloud service (UUID = 73a0a571-e48a-4ccf-af64-5342020b05a0, base URL = http://localhost:8080)
(Feel free to mark this post as a solution for others who also have this issue)