I believe you're referring to the fact that a person baptized needs a "Christian Name" and that name would be their protection throughout their life.
The sacrament of Baptism is conferred "in the name of the Father and of the Son and of the Holy Spirit." In Baptism, the Lord's name sanctifies man, and the Christian receives his name in the Church. This can be the name of a saint, that is, of a disciple who has lived a life of exemplary fidelity to the Lord. The patron saint provides a model of charity; we are assured of his intercession. The "baptismal name" can also express a Christian mystery or Christian virtue. "Parents, sponsors, and the pastor are to see that a name is not given which is foreign to Christian sentiment
When your saint's "name day" comes around, it's a bit like your own spiritual birthday, etc.. It's a good an pious practice hat should probably be done more. This is usually your own name, unless your parents named you something that's not particularly Christian and then you get a new name at Baptism.
The code of Catholic Church law issued in 1918 instructed pastors to see to that each person baptized was given a “Christian” name. If the parents of a child to be christened insisted on an “un-Christian” name, the baptizing priest was instructed to add a Christian name on his own initiative and to enter both names in the baptismal register.
The revised code of Church law (1983) is less restrictive. It simply forbids the giving of a baptismal name that is “alien to Christian sentiment.” Most likely, however, this broader rule will not have much influence on the established custom of preferring the names of saints. This is good, for it is better for us to have a heavenly patron saint whom we can admire and turn to. - Baptismal Saints
However, I don't think it's universally recognized that you can only have one patron, especially since that patron was probably chosen for you by your parents and most likely doesn't have a whole heck of a lot to do with your vocation or particular needs. St. Theresa of Avila, who was a bit of a rule breaker - so maybe this isn't a good example, took St. Joseph as her patron to help her in her work.
and I took for my patron and lord the glorious St. Joseph, and recommended myself earnestly to him.
Life of Teresa of Jesus
She believed that he could help her with everything.
I took the name Alphonsus, because I liked Frank Zappa and that was the closest I could get to St. Alfonso (of pancake breakfast fame). But he did guide me quickly back into the Church when I bought a big book from Ligouri press on a whim that turned out to be a Catechism (which I don't recall having seen in Catechism class).
Digressions aside, I believe you must have several saints that you are devoted to in order to survive as a Catholic and I don't believe that one will be saddened by your veneration of another.
St. Teresa heard from the next highest authority too, telling us in her autobiography, “She (our Blessed Mother) said that I pleased her very much by being devout to the glorious St. Joseph.”
National Catholic Register
Lastly, and this is just a practice that may be applicable to my diocese, given the way we structure baptisms and confirmations, but when confirmation candidates in High School prepare for the sacrament, they're tasked with finding a saint to go and make their confirmation patron. It's never incumbent upon them to choose their Baptismal name, although that might be a lapsed practice that uninformed catechists, like myself, would be better off correcting.