I see the green light on the dashboard that says sport but don’t know what it is and whether it should be on all the time. I also don’t know where the button is to turn it off either. This car is secondhand for me and done 291,000 kms🙏🏻😊
At least for the 2005 model the following might be useful: http://www.subaruoutback.org/forums/109-gen-3-2005-2009/411177-new-me-2005-sport-light-cluster.html
Once the shifter is in D, you can slide it to the side to engage sport mode. This does two things: increase the sensitivity of the first half of the accelerator pedal and change the shift timing to allow higher RPM for better acceleration. You don't get any extra power, but the change in the computer's interpretation of the accelerator input really does make the car feel faster. It is also harder to get advertised MPG in that mode.
Are you sure you are not sliding the shifter to the side from the D mode?
My car, a 2016 Toyota RAV4 hybrid also has the same "S" mode in the automatic (or well, actually, hybrid eCVT) transmission that allows manual emulated shifting. When I went for the first time into a test drive in that car, I accidentally turned the shifter all the way to the "S" position, which meant I probably got slightly lower MPG than possible. Nowadays, after purchasing the car, I use the "S" position only whenever I need the extra acceleration. Otherwise I use "D".