I doubt that many people at 16/17/18 have ever said that, but after the end of their time in whatever service that requires an oath to be sworn a lot of people do think a variation of that - perhaps not pride at serving the Queen personally, but pride at serving the same thing that she served for so many years. She was (and the monarch is) a part in the same machine as them, after all in a way that politicians very rarely are (and of course there was the massive difference between a visit to your base / ship / station / whatever from HM compared to any politician).