It's tradition because many of the deep-level lines were built by a company owned by Charles Tyson Yerkes, who bought up companies that had obtained Parliamentary permission to build them but didn't have the money. This company also electrified some of the sub-surface lines. So it's the system, not the trains, that was built by an American company- though most of Yerkes's trains were built by the American Car & Foundry Company. The Underground also refers to places where trains can move between tracks as "switches" as in the US rather than "points" as in the rest of Britain.
As for "car" versus "carriage" today, it's a mixture. Two commonly-heard announcements on the Tube are:
Please move right down inside the cars, use all available space
Upon arrival, the last set of doors will not open. Customers in the last carriage, please move towards the front doors to leave the train.