Yup- although plenty of cars with automatic gearboxes do have stop/start. I know Volkswagen group cars with the DSG gearboxes with stop/start have quite a good system. If you press hard on the brake pedal then the stop/start kills the engine, if you only press it hard enough to hold the car stationary, then the engine stays on. Works well. I've not driven any other vehicles with an automatic and stop/start so can't comment on other manufacturer's implementation.
Sorry going miles off topic here, but cars etc are my day job, as are emissions...
The DSG is not a traditional auto it’s 2 automated manuals working in tandem in very simple terms. This is why it can easily be “stop started”
As for saving fuel...in theory it does, but only if you have extended periods of idling in your journey... and the flip side is the constant starting drains the battery (stop start cars tend to have larger batteries), now modern cars have smart alternators which will load shed, charge more when in gear coasting, etc but there comes a point where the battery has to be charged and therefore the engine has to run...
Then you get into the longer term issues, bottom bearings, turbos, starter motors, dual mass fly wheels all take a battering from stop start, where as keeping things running avoids a lot of these issues... they are unlikely to fail within the warranty period but they will fail sooner than they used to...
The last thing I will say is this, the old NEDC emissions cycle has a lot of idle time in it...that is why stop start gives such a big gain on paper... in the newer WLTP drive cycle there is far less idle and the fuel con savings come from better calibration and smarter tech, to the point that I expect at some point if it wasn’t seen as a backwards step by the public some manufacturers would happily ditch stop start to save costs on components etc.