Most, if not all, cab drivers cannot take possession of items left in the cab. New York, for example, says: §54-18(a) Inspection Following Trips from Airports.

Most, if not all, cab drivers cannot take possession of items left in the cab. New York, for example, says: §54-18(a) Inspection Following Trips from Airports. Immediately after completing a trip to Kennedy, La Guardia or Newark Airports, the Driver must inspect the interior of the Taxicab and the trunk compartment, if used, to ensure that Passengers have collected their property. §54-18(b) Handling of Lost Property. Property found by a Driver in a Taxicab must be returned to the Passenger if possible; otherwise, it must be taken without delay to the police precinct closest to where the Passenger was discharged. §54-18(c) Notifying Commission Regarding Lost Property. If the property is not returned to the Passenger, the Driver must promptly inform the Commission of the details regarding the found property and the police precinct where it is held. Note that 54-18(b) is not inclusive of (a), but only when transporting passengers to the listed airports is an inspection of the passenger area for lost or misplaced items required. If any items are found outside of those conditions, they still must be taken to the closest precinct, and the Taxi commission must be notified of the property and to which precinct it is located. It is not the responsibility of the cab driver to track you down, outside of yelling at you while you are walking away from the cab. After that, it is your responsibility as a passenger with a lost item to track it down.