"To book this seat" means to reserve or secure the availability of a particular seat or place, typically in a transportation vehicle or at an event, by making a reservation or purchasing a ticket in advance. Full definition
There seems to be such confusion that if you were to book a seat on a cheap flight that it'll automatically be the worst flight you've ever faced. (comox-valley-tourism.ca)
The letter writers had not realized that they now had to book seats in advance. (canadianbusiness.com)
If she doesn't book a seat for the baby, she will have to gate - check the car seat. (havebabywilltravel.com)