In a manual transmission car, the transmission control is generally called the "stick shift". What is it called in an automatic car?
Both are just called either the gear lever or shift lever (in the UK anyway) Auto or manual then just refers to the transmission type.
I'm in the Midwest region in USA, I've heard "gearshift" and "gear selector" used for both manual and automatic, "shift lever" for an automatic with a stick, or "shift knob" for a dial-type selector.
Gear knob, shift knob. Depends on the shape. But the UK defined terminology is correct. In USA it's just referred as gear knob.
As per the other answers, it is called the gear lever, selector, knob, etc...
If you are trying to find the name of the manual shifting in automatic, then the portmanteau term is "Manumatic". However, I have mostly heard it as TipTronic, S-Tronic, E-Shift and SportShift trademarks from different manufacturers.