During landing, are passengers allowed to keep plastic cups served to them?
There is a regulation that specifically addresses this situation –
121.577 Stowage of food, beverage, and passenger service equipment during airplane movement on the surface, takeoff, and landing.
To make the regulation easier to understand, I make bold the words that address this issue.
(a) No certificate holder may move an airplane on the surface, take off, or land when any food, beverage, or tableware furnished by the certificate holder is located at any passenger seat.
Based upon the regulation, passengers that are consuming a beverage provided by the airline should finish it and pass the cup to a flight attendant during collection. Everyone should understand that this does not mean flight attendants should call the captain because a passenger has a plastic cup at their seat and to abort the landing. If that were to ever happen, it could be viewed as perhaps the most unjustifiable reason to abort a landing. Safety is not compromised, so the flight crew should never be involved in anything related to a service item. The only exception could possibly be if for some reason service carts were unable to be stowed back into the galley. Even with that, this type of problem existed mostly with the older aircraft which had mushroom floor anchors to secure the cart within the galley.
Flight attendants are required to collect service items prior to landing and do so per regulation and crewmember training procedures. Most passengers would comply with this request when provided a few minutes advance notice that flight attendants will be coming through the aisles to collect all remaining service items. This would give them an opportunity to finish their drink and be ready for the flight attendants when they come around for trash collection.