What is this kind of switch called?
A datasheet for a similar switch is shown here.

This datasheet refers to the switch as a "push operated" type of "thumbwheel switch."

You may also hear this called a BCD switch, since the selected number (0-9) is provided, electrically, as four bits coded in BCD (Binary Coded Decimal) format.  If the switch includes all 16 of the four bit values (0-F) is may be referred to as a HEX switch.