Long story short: the Hundred Years War was pretty much just a protracted civil war fought by interlinked dynasties ruling over domains on both sides of the Channel, and the French crown itself was involved in that claim game.

France and England actually came very close to being united together: the last phase of the "war" (actually a succession of conflicts) involved Charles VI marrying his daugther to the king of England, and adding clauses in his will to set aside his son (Charles VII) and bar him from inheriting the French crown, which would have tied, eventually, both realms together.

That obviously didn't go well, and the will was nullified under two main rationales: Charles VI was notoriously subject to mental troubles and crises, and Charles VII's administration and legists dugged out old Frankish laws to justify preventing a woman from transfering the French crown.

Follows the last part of the "war", that's Joan of Arc, Burgundy switching sides, siege of Orléans, etc. Interestingly, English/British monarchs only relinquished their claim on the French crown under Georges III, somewhere by 1800 iirc.