Well, to be an official patron saint, the Holy Father will declare that a specific saint is patron of something, but there are also unofficial patrons that evolved from popular devotion. In the case of patron saints of cities or countries, quite often the patron saint was born/buried or lived/died there, but as I said, this is not always the case and patronage can be based on popular devotion (e.g. St. Joseph being the patron saint of Canada, even though there are canonized Canadian saints and martyrs. Actually, these saints and martyrs were well known for their devotion to St. Joseph!)
You can ask any saint to intercede for you in whatever you like, but that on its own doesn’t make them the patron of that. If you were baptized/confirmed with the name of a saint, then they are your special “patron” saint(s), and you share a connection with them in name. You are correct that often saints do have a common experience in life with whatever they are patron of (such as St. Joseph of Cupertino who would levitate/fly through the air and aviators), but this is not always the case (e.g., St. Gerard Majella is called the patron saint of the falsely accused since he was falsely accused, but he is also popularly acclaimed the patron saint of mothers despite the fact that he was obviously never pregnant/trying to conceive. Usually this title is attributed him for the miracle he obtained for a mother during his lifetime.)
I hope this helps. God bless you.