To unlock the Sidecar Sidekick trophy in Far Cry New Dawn, you will first need to complete the Man Eay Dog World side mission to recruit Timber as a Guns for Hire. You can find the side mission at Meatfort, which is east of the Broken Forge Outpost.
Once you have completed the side mission and recruited Timber, go to a garage and craft the "Rusty Sidecar". Press Triangle to spawn the "Rusty Sidecar" at a captured Outpost (you can't spawn vehicles at the garage in Prosperity) and assign Timber to your roster so that he follows you. Get in the "Rusty Sidecar" and once Timber jumps in you can begin to drive around.
It doesn't have to be five minutes of continuous driving. It is the total time combined across as many different journeys as required.