What is this control that is used to select an item in this screenshot called?
This is also in the latest Facebook for Android update. What is it called?
Apple calls it a "Picker" in its iOS Human interface guidelines.
A picker displays a set of values from which a user picks one.
To learn more about defining a picker in your code, see UIPickerView Class Reference. Appearance and Behavior
A picker is a generic version of the date and time picker. As with a date and time picker, users spin the wheel (or wheels) of a picker until the value they want appears. The overall size of a picker, including its background, is fixed at the same size as the keyboard on iPhone. (For more information about the date and time picker, see “Date and Time Picker.”)
If you are specing out a responsive design that covers multiple devices, you may want to consider using "native device controls". What we would use on the desktop is a derivation of a select list; this describes the fundamental user interaction with this element.
This does not mean "this will make every thing work like the Apple Picker", it means this will use the device default. Just be careful when laying down the requirements, you want to make it as unambigious as possible.