For at least 1 major carrier, passengers are given a voucher with a dollar amount: the voucher may read 'good on any cab for $15'.
Occasionally after irrops you would see cabbies at the airport looking for a cashier or office where they could cash-in their voucher. Vouchers were not paid at the airport but went thru a specific department. I assume the cabbie eventually gave them to their company and the company submitted them. I do know the cabbies claimed they did not get paid until the carrier paid and this could take severl weeks.
Cabbies did not like getting the vouchers.