The Greek roots of the term “Catholic” mean “according to (kata-) the whole (holos),” or more colloquially, “universal.” At the beginning of the second century, we find in the letters of Ignatius the first surviving use of the term “Catholic” in reference to the Church. At that time, or shortly thereafter, it was used to refer to a single, visible communion, separate from others.
The term “Catholic” is in the Apostles’, Nicene, and Athanasian creeds, and many Protestants, claiming the term for themselves, give it a meaning that is unsupported historically, ignoring the term’s use at the time the creeds were written.
Early Church historian J. N. D. Kelly, a Protestant, writes: "As regards ‘Catholic,’ its original meaning was ‘universal’ or ‘general.’ . . . in the latter half of the second century at latest, we find it conveying the suggestion that the Catholic is the true Church as distinct from heretical congregations (cf., e.g., Muratorian Canon). . . .
catholic.com/library/What_Catholic_Means.asp
I would like to quote Pope Benedict XVI from his book “Called to Communion”:
"The institution of the most holy Eucharist on the evening before the Passion cannot be regarded as some more or less isolated cultic transaction. It is the making of a covenant and, as such, is the concrete foundation of the new people: the people comes into being through its covenant relation to God. We could also say that by his eucharistic action, Jesus draws the discipline into his relationship with God and, therefore, into his mission, which aims to reach “the many”, the humanity of all places and of all times.
These disciples become a “people” through communion with the Body and Blood of Jesus which is simultaneously communion with God. . . . This fact already implies that the Church and her unity are but one. ** It is true that the many celebrations in which one Eucharist will be realized also point ahead to the multiformity of the one Body. Nevertheless, it is clear that these many celebrations cannot stand side by side as autonomous, mutually independent entitites but are always smply the presence of one and the same mystery.**
Those that make up the Church (not churches) are God’s elect and we are one as the body of Christ is one, therefore, to say that on the local level we are whole is wrong. We all share in the same mystery (eucharist) and therefore are united in this mystery as one, i.e., we are all in (one) communion with Christ’s Body (which is one) and we cannot be separated.