Hmm I'll take your word for it that that's true, but I would say the examples are very different. Independence Day is a title/holiday retroactively created to commemorate the event (which apparently might not have even happened on July 4).

Whereas D-Day was something soldiers used to describe that specific day even before it happened. And you would hear things like "D-Day plus 23" to describe points in time, you wouldn't have to specify the year

So to me the Independence Day analogy is a little weak.

That was the original usage but there is no reason to think usage hasn't changed in Britain, if a British person is saying it is now used this way.