I think they are claiming that the watch was a Spitfire pilot's watch, not that the watch is actually called a 'Spitfire'.
Oh don't be so cynical, it's a well known fact that the commander of Twelve Group, fighter command, Trafford Leigh-Mallory insisted that all of his pilots wore Rolex Air King watches to commemorate the successful ascent, but tragically unsuccessful descent, of Everest by his brother George Leigh-Mallory, who carried a Rolex Oyster to the summit as part of the top secret R&D process that led to the Explorer. The dispute between Keith Park, AOC commanding Eleven Group, who insisted his Spitfire pilots wore the almost eponymous Omega and Mallory who was equally insistent on them using the Air king led to the so called 'big king' controversy, one of the most contentious events of the Battle of Britain.
The famous ace, Douglas Bader, argued that the allegedly superior timekeeping of the Air King allowed his pilots to rapidly form up into larger formations, the so called 'big kings' named after the watch that he, and his commander, believed made assembling such complex formations possible. Park, on the other hand, insisted on the flexibility of using single squadrons. Whether this strategy was forced on him because of the reliability of his watches or a tactical choice has been a moot point for over seventy years. Likewise, the assertion that the Air King had a tendency to run slow, causing the Big Kings to tend to intercept aircraft after they had bombed rather than before has been hotly denied by the secret Rolex service department at White Waltham.
Whatever the truth, the fact is that, shortly after the successful defence of British airspace that led directly to the cancellation of Operation Sealion, the proposed invasion of Britain, the C in C of Fighter command and architect of the victory was retired and shunted abroad, while Park, the commander who bore the brunt of the battle, was moved to a training command.
While this looked like a victory for Rolex, they rapidly realised that the market was about to be flooded with cheap American fakes and abruptly changed their marketing strategy; Bader, their most successful advocate, was parachuted into Germany (under a cover story of being shot down) to set up a successful chain of Rolex concessions in prisoner of war camps. The rest, they say, is history.
Sorry.