“Around the year A.D. 107, a bishop, St. Ignatius of Antioch in the Near East, was arrested, brought to Rome by armed guards and eventually martyred there in the arena. In a farewell letter which this early bishop and martyr wrote to his fellow Christians in Smyrna (today Izmir in modern Turkey), he made the first written mention in history of “the Catholic Church.”
All Christians believe in the Universal Church. We just don’t limit it to one single church in Rome...
And there's the problem ... Is there to be only one Church or many? According to Scripture, Christ wanted us to be one (John 17:22-23). We are all as a Church to be of one mind and to think the same (Philippians 2:2; Romans 15:5). There is only to be one "faith" (Ephesians 4:3-6), not many. For the Church is Christ's Body and Christ only had one Body, not many. Also, since the Church is Christ's Bride (Ephesians 5:29), can Christ be married to more than one wife (essentially a spiritual form of the the sin of polygamy)? No, Christ can only have one wife (i.e., one Church, not many).