Dual passports with different names, and names on bookings
A proportion of people have dual nationality, and it is common for the two passports to be in different names, not least when a woman from country A gets married and now lives in country B under her married name.
So which name do you use when making a reservation, which will be checked against your passport ? You may think either could be used, as long as you use the same passport in both directions.
But what if travellers between the two countries require visas, which are also checked at the start of the trip. Countries will never give visas to somebody who is also one of their own nationals. For both sectors you will be need to use a different passport, that of the destination country, to show eligibility to enter that country without a visa. One of these will not match the booking.