Hi Vanessa: I don't want to come across as patronising but have you read the manual? The on off button for the Start / Stop is in the row of buttons above the Climate controls, Mine is to the left of the Parking Assist Button.
My A6 is a '16 Automatic model and the start stop does not function every time. If you stop at a junction and it is in a condition to stop the engine then you get a green icon on the dash between the dials and the rev counter drops to "ready" (not 0). If stop start is not functioning then the icon between the dash dials will have a strikethrough and be white.
When the engine stops look at the rev counter - does it drop to ready? You mentioning pressing the Start button twice would suggest that the car has stopped the engine purposefully and on your first press of the start stop button you have actually "turned of the ignition" with the second press restarting the engine.
Is your car auto or manual? With a manual A5 Cab I had on loan it would only stop start when out of gear with clutch released. See link below...
https://www.audi.co.uk/glossary/s/start-stop-system.html