Captain America: The First Avenger was very much a love letter to the gung-ho, patriotic roots of the Star-Spangled Avenger, and even though Cap has become increasingly jaded in films like Captain America: The Winter Soldier and Captain America: Civil War, he remains an idealist who embodies American values in a meaningful way.
If any character could have a birthday on the 4th of July, then it would have to be the First Avenger.
It honestly makes perfect sense to see a character like Captain America have such an on-the-nose birthday in the Marvel universe, particularly when we examine the nature of the hero throughout his tenure in the Marvel Cinematic Universe.