No, that’s definitely not the case. This is not an Android device, mobile data is disabled. I use the MobileID on the Swisscom SIM card, not the app!
The terms of use state: “To use Mobile ID, an active connection to the telephony and SMS services in your mobile service provider’s network is required.”
There is nothing about data connections.
I check the account balance before requesting a MobileID and after requesting a MobileID and my account balance empties. If I don’t use MobileID, my account balance remains the same, so no other application uses the mobile data. I’m also constantly on WiFi.