I didn’t notice any of that, I was just looking at the rug:
Joke’s on me.
Well, isn’t there a standing White House set in Hollywood that many productions have used over the years? Oliver Stone’s Nixon came out the year before ID4, so maybe they just used the same carpet.
JEDIT: I was right!
http://www.nytimes.com/1997/04/13/movies/oval-offices-by-way-of-hollywood.html