I think it’s a combination of factors.
The Holy See declares a patron saint (St. Joseph the Worker, St. Thomas Aquinas, patron of universities …) and that sometimes assigned to a feast day.
I think the old saints were considered patrons just based on venerable custom. Or perhaps the local bishops declared them patrons.
It’s a good question. The first thing saints are patrons of are churches given to their name.
Well formally these days, the decision is made by the Vatican. However, in the past, I suspect it had a lot to do with popular acclimation similar to the way that it could result in sainthood. Thus, I doubt that the Church ever needed to proclaim St. Patrick the Patron of Ireland, rather it simply recognized him long after the Irish had .
DISCLAIMER: Catholic Answers has turned over the archive to Catholic-Questions.org and no longer owns, manages, or moderates the forums. For additional apologetics resources please visit www.catholic.com.