Malory mentions that this is "not my first Grand Prix". This is an inside joke, as Jessica Walter, who voices Malory, previously starred in the racing film Grand Prix (1966).
In the opening race, the three racers are named are Bell, Bivins, and DeVoe. This is an homage to Ricky Bell, Michael Bivins, and Ronnie DeVoe of the musical group Bell Biv DeVoe.
The second time the Jackie Stewart-style announcer calls the race, the drivers' names are Moonsie (Susan Moonsie), Bennett (Brenda Bennett), and Kotero (Apollonia Kotero), all the last names of members of Apollonia 6, a girl group formed by the musician Prince in the early '80s, though not all three were in the group at the same time.
When the concierge preemptively supplies Archer with chips, he replies: "Thanks, Radar." This is a reference to the character Corporal Radar O'Reilly, played by Gary Burghoff in both M*A*S*H (1970) and M*A*S*H (1972), which were drama/comedies set at the 4077th, a fictional mobile Army hospital, during the Korean War. Radar was the company clerk to the commanding officer, and he had an inexplicable knack for predicting/anticipating things seconds before they happen.
"Monegasque" is a French term referring either to the French dialect used in Monaco, or something in or related to Monaco. The title roughly translates to "The Monaco Game".