What is the spiritual significance of a patron saint?
The spiritual significance of having a patron saint is quite simple. We have a saint in heaven to, as one can say, have as a heavenly friend that will pray for us, our work and our spiritual needs.
The Church encourages us to develop spiritual friendships with those saints more closely united to us and known as Patron Saints. The reason why certain saints become patron saints of particular persons, associations or countries.
Apart from having a group of faithful that invoke a particular saint for for a particular cause, there are no official rules or norms in order to establish a certain Catholic saint the patron saint of something in particular.
In other words, the faithful remain free to add more “patronages” to particular saints as their own particular devotion, needs or circumstances deem it necessary.
On very, very rare occasions, the Holy See will name someone a patron saint for of certain cause or people. This actually happened to St. Thomas More:
On 31 October 2000 Pope John Paul II declared More "the heavenly Patron of Statesmen and Politicians". More is the patron of the German Catholic youth organisation Katholische Junge Gemeinde. - Thomas More
Catholic piety and Church teaching go hand in hand here.
There is no particular manner or special way to talk with our friends in heaven. The Church leaves it to our personal preferences in regards to our piety devotions with the saints. We are free to develop our friendship with those saints close to our hearts.
On another note saying novenas towards the saints is very powerful. There are literally hundreds of novenas. One of my favourites is the Novena of Grace in Honour of Saint Francis Xavier. The Novena of Grace is quite famous.
Yet another form of devotional prayer to the saints comes in the form of litanies!
On a personal note: There is no patron saint for recycling or those involved in recycling. I do a lot of recycling, but for years I did not have a patron saint for it. Then one day, I came across the following article: Relic of St Clement found in trash settles into Westminster Cathedral. Ever since that day, Pope St. Clement became my patron saint when I am recycling. There are no rules or norms forbidding my choice in patron saint!