When the push button won't start the car
A push-to-start vehicle checks for a recognized key before it allows the start. When that fails, the key is only one possible cause. A dead fob battery, a weak 12-volt vehicle battery, the brake pedal not being fully pressed, a steering lock, the shifter position or a vehicle fault can produce similar symptoms. I ask about these first, because a new key won't fix a problem that isn't in the key.
Recognition versus registering a new key
If a working key is simply not being detected, a fob battery or the backup start position described in your owner's manual may get you going. If the key is lost, damaged or never registered, a compatible key has to be registered to the vehicle so it is recognized at the start button. That is a different job from pairing the lock and unlock buttons, and support depends on the vehicle.
Push-button systems vary
Not every push-button vehicle uses the same backup start method, the same number of cabin antennas or the same registration procedure, even within one brand. For how proximity keys work more generally, see smart keys. For a vehicle with no working key, see lost car keys.