Technically: Thats no app but normal messaging

The number behind is not the real phone number from the boss, but a forwarding number. If you send a message to the number, the message gets in a queue for the boss to read and he will see the message when he is back in office.

Actually: While these kinds of services exist, they would result in a green message box in iMessage. So here it's clearly just a joke between pals

I have chosen to believe they type that manually every time.

A lot of phones have built in car mode settings that block notifications and send out prewritten responses to tell the messenger they're currently driving.