Again, a mobile phone works regardless of where you purchase it. Your banking systems, etc are not authorised to a mobile phone, though. They're authorised to the one mobile phone that was authenticated.

It's not enough to buy a new phone. You need to contact every single service that's bound to the lost/stolen/broken phone and authorise the new one before you can use their services.

In the case of the car, it's enough to just fill up the tank with petrol purchased anywhere. It's ready to carry on at that point.