Well what does antiquity say about the name, but tell of the same phenomenon that exists today.
“For in the Catholic Church, not to speak of the purest wisdom, to the knowledge of which a few spiritual, men attain in this life…–not to speak of this wisdom, which you do not believe to be in the Catholic Church, there are many other things which most justly keep me in her bosom. The consent of peoples and nations…so does her authority…the succession of priests…[a]nd so, lastly, does the name itself of Catholic, which, not without reason, amid so many heresies, the Church has thus retained; so that, though all heretics wish to be called Catholics, yet when a stranger asks where the Catholic Church meets, no heretic will venture to point to his own chapel or house. Such then in number and importance are the precious ties belonging to the Christian name which keep a believer in the Catholic Church…” Augustine (397 AD)
“The Article, In one Holy Catholic Church,’ on which, though one might say many things, we will speak but briefly. It is called Catholic then because it extends over all the world, from one end of the earth to the other; and because it teaches universally and completely one and all the doctrines which ought to come to men’s knowledge, concerning things both visible and invisible, heavenly and earthly… for this cause the Faith has securely delivered to thee now the Article, And in one Holy Catholic Church;’ that thou mayest avoid their wretched meetings, and ever abide with the Holy Church Catholic in which thou wast regenerated. And if ever thou art sojourning in cities, inquire not simply where the Lord’s House is (for the other sects of the profane also attempt to call their own dens houses of the Lord), nor merely where the Church is, but where is the Catholic Church. For this is the peculiar name of this Holy Church, the mother of us all, which is the spouse of our Lord Jesus Christ, the Only-begotten Son of God.” Cyril of Jerusalem, Catechetical Lectures, 18:23,26 (A.D. 350).
We can test this same phenomenon today, by asking anyone you meet in any place, where the Catholic Church is and see where it is they direct you.
The Catholic Church today is the same Church as these fathers, with the same faith and belief.
“We must hold to the Christian religion and to communication in her Church, which is Catholic and which is called Catholic not only by her own members but even by all her enemies. For when heretics or the adherents of schisms talk about her, not among themselves but with strangers, willy-nilly they call her nothing else but Catholic. For they will not be understood unless they distinguish her by this name which the whole world employs in her regard.” Augustine, The True Religion, 7:12 (A.D. 390).