I don't think Norman and French identities in this context should be viewed as in opposition. Normandy was part of France (although its dukes becoming kings of England would later complicate that status) but it was not the entirety of France. It had distinct regional customs, it had its own dialect, it had its own army (although hardly unique regarding French regions in that era) that did the conquering, and it had its own lords who by and large (with IIRC a couple Breton or Flemish exceptions) became England's ruling class once the conquest was completed.

JFK was from Massachusetts. Massachusetts is part of the United States. Yet (to stretch this metaphor further) if JFK had been governor of Massachusetts, and led a successful invasion of Canada with the state militia, people would talk about Massachusetts' conquest of Canada, not America's, and perhaps the fact that many Massachusetts residents, including the governor at the time of conquest, were Irish-Americans would play a disproportionate role in the historiography. And this distinction between governor and president shouldn't be overlooked, unlike JFK, who was president of the whole USA, William of Normandy was never king of France.

Calling William a Norman is more specific than calling him French, and given the role of that French province in the conquest, it's a specification that's often used. But that shouldn't be read to imply that he wasn't also French.

In 911 Rollo signed the Treaty of Saint-Clair-sur-Epte with King Charles III of France which allowed Normans to settle in Neustria. This was after invading Paris, but it granted the lands we will eventually call Normandy. At the time, it was known as Rouen. Rollo was given the title Count of Rouen. His successor, William I, expanded Rouen. However, many consider him a Duke of Normandy despite the title not technically existing until the 11th century.

William I (Longsword) attacked Arnulf I, Count of Flanders in 939. He, along with with Louis XIV, King of France, retaliated against William I where he pledged loyalty to the King of France. He was killed in an ambush in 942.

That same year, Louis XIV IV seized Rouen and took over the lands. Richard I, son of William I, was only 10 years old at the time. He was taken under the care by Count of Ponthie, but managed to ally himself with the Norman and Viking leaders. By 947 he claimed Rouen back under his rule.

Richard II (Richard The Good) took over rule in 996 when his father died. He married his sister (Emma of Normandy) to King Ethelred of England (important). He helped Robert II of France fight off the Dutchy of Burgandy, and later forged an alliance with Brittany with the marriage to Judith of Brittany.

Richard III took over after the passing of Richard II, but it lasted a short year (1026 - 1027). His younger brother, Robert I, was next in line. This guy alone is an interesting story in himself as it's believed he was responsible for his brother's dead. Earlier, he attempted a civil war against his brother. He also assembled an army against his Uncle, caused disturbances in the civil war in Flanders, etc. He made his illegitimate son, William, heir while he went to Jerusalem. He died in 1035 probably from illness.

Remember Judith? Yeah, her Grandson is William The Conqueror. Remember her husband? King of England (Æthelred). They were driven out of England by Denmark and went to Normandy for refuge in 1013. This is where William was born in 1028 by Judith's daughter Herleva and Robert the I. This is part of purpose behind William's conquest of England.

The Archbishop Robert and King Henry I of France mentored William. When the Archbishop died, Normandy went into anarchy. In 1046, with the help of Henry I, William led a rebellion in lower Normandy and eventually reclaimed all of the land back. The rest is history.

During this whole time, however, the people in Couen had always considered themselves as Norman. Even though it was fiefdom of France, it was essentially considered a vassal where Normans could reside. The term Vassal wasn't used until about 1057. I think this is where the confusion is. I think if you could relate it to modern times, it would be like Puerto Rico to America. It is technically a part of America, but they're still their own territory and government, but America allies with them to help when necessary. They don't consider themselves American, rather Puerto Rican.

One thing I didn't cover is when Normandy first started getting called Normandy, and that was about the time Richard II and is considered the first Duchy of Normandy.

Sources:

Eleanor Searle, Predatory Kinship and the Creation of Norman Power, 840–1066 (University of California Press, Berkeley, 1988)

Bates, David R. William the conqueror. The History Press, 2004.

François Neveux, A Brief History of The Normans (Constable and Robinson, 2008)