The Mossad did something dumb like that once. Neckties are not widely worn in Israel, so the photo office that made fake passports had some neckties on hand when taking pictures. But only three different neckties. After a while, some other countries caught on.
Why not. Remember Elite Russian GRU hackers identified because they all registered their cars with the DMV at the same address? https://www.bellingcat.com/news/uk-and-europe/2020/10/22/rus...
Smart people do dumb things.