Drivers licences in Australia are the de-facto ID. Including proof of age to purchase cigarettes/alcohol, many bars scan IDs as you enter in case you cause damage or start a fight. To proving identity for a variety of things including (but not limited to) new phone plans, bank accounts, etc.
Most Aussies carry their drivers licence with them everywhere.
Regarding the stationary part. Though it varies state by state, my state (Victoria) considers it an offence to use your phone while stationary. If the vehicle is legally parked and engine off you're fine. This is to cover those using their phone while stuck in traffic.