The problem being that what is meant by the term “the Catholic Church” has itself changed over the course of 2000 years of history. Today what is meant is what is known to some as the Roman Catholic Church, a institutional organization with its seat at Rome. But when Ignatius spoke of the Catholic Church he meant something different than that which is meant today:
It was saying that no matter where one was in the world, it didn’t matter what country, what ethnicity, what bishop was presiding, that there was just one church that was universal and all Christians are one in it, no matter what other difference there might exist between them. Iraneus would put it like this: This is what it means to be “catholic”, this is the “catholic Church” (small “c” on catholic, large “C” on Church) spoken of in the apostles creed. And this catholic or universal church still exists, not as headed by the Pope in Rome, but as headed by Christ in the hearts and minds of those who belong to him every local congregation and no congregation scattered around the globe.
So, how does anyone answer when the Catholic Church came into existence? I don’t think you can point to a precise time. The Catholic Church as the term is used today was NOT present when Jesus established the Church on Peter. I don’t think it was yet present with the passing of the apostolic age. I don’t think it was even present at the time of the writings of Ignatius and Iranesous, though you can see some of the things that would become foundation stones used by the Catholic Church in their writings. Yet, by the time of Leo IX and the Great Schism it was clearly institutionalized. I think one could make an argument for the transition occurring sometime around the time of Leo I, but there isn’t going to be any hard and fast date.
The Catholic Church evolved over time. It is like asking when did the wolf become a dog. There was a time when there was a canine that was not a dog that eventually became a dog. And there was a time when there was a catholic Church that was not the Catholic Church. And sometimes between then and now both evolved. But to put a date on it, and say that the first dog was born on this date or that the Catholic Church began on that date, it is an impossible question that one could not have answered even if you had been there. But just as it is clear that a dog is descended from a wolf but is not a wolf, so to it is clear that the Catholic Church is descended from the catholic Church but is not the same as the catholic Church spoken of by Ignatious.