- Go to IMDB and look up movies with Castle Rock Entertainment attributed to it
https://www.imdb.com/search/title/?companies=co0040620
- Number 4 is the American President, and if you go to page 4 of the pictures, you'll find this:
https://www.imdb.com/title/tt0112346/mediaviewer/rm3122897409/
-Rug looks awfully familiar, doesn't it?
https://www.bostonglobe.com/2021/01/21/nation/photos-biden-brings-slightly-new-look-oval-office/
-Now one thing I did find interesting when looking at Clinton, Bush, Obama, and Trump's oval office decor, Clinton too had the same rug.
http://www.americanheritage1.com/assets/images/presidents/bill-clinton-oval-office-sofa-1.jpg
-Elvis vs Nixon also had a similar Oval Office, VERY SIMILAR https://www.rollingstone.com/wp-content/uploads/2018/06/rs-237398-MCDELAN_EC032_H.jpg
-Confirmation that Castle Rock Pictures has a Oval Office set per a 1997 NYTimes article
https://www.nytimes.com/1997/04/13/movies/oval-offices-by-way-of-hollywood.html
-Since it was in Independence day as well, went and looked for it in IMDB, here's a photo
https://www.imdb.com/title/tt0116629/mediaviewer/rm2294328065/
-And here it is in the Nixon movie from 1995
https://www.imdb.com/title/tt0113987/mediaviewer/rm756626944/
EDIT: As LurkerWill pointed out, the rugs are very different, but the sets are similar. The most similar set goes to Clintons, then The American President, and then Elvis vs Nixon
I do think the presidential set is a fun theory, but I'm not really sure why the rug is so important. The only rug that was similar was Clinton's.