When babies are born, sometimes they aren't named for a day or two (depending on the culture and circumstances). When the baby is given an official name, he/she does not become a different person. They are the same person with a name (label) to be able to identify them.
The Church founded by Christ very early on came to be known as the Catholic Church. St. Ignatius of Antioch, the bishop of Antioch ordained by St. Peter, was captured by the Romans. While they were transporting him to be martyred for the faith, he wrote a letter to the Smyrnaeans around 107-110 A.D., referring to the "Catholic Church," not in such a manner as if he were coining the term, but in such a manner in which he fully expected the Smyrnaeans to understand what he was talking about. It says in paragraph 8, "Where the bishop is present, there let the congregation gather, just as where Jesus Christ is, there is the Catholic Church."
See the entire letter here:
https://www.orderofstignatius.org/files/Letters/Ignatius_to_Smyrnaeans.pdf
The buildings of the Church aren't an integral part of the real Church, but a tool thereof. The doctrines of the Church were handed to the Church by Jesus Christ, Who gave them to the Apostles, and who, in turn, handed them on to their successors, the bishops, who have done likewise for 2000 years now. These original doctrines are the doctrines of the Catholic Church. The Church has never changed any of the doctrines given her by Christ, nor does it claim the authority to change them. Christ promised to remain with His Church until the end of time (Matt. 28:20). This means He will protect His Church from teaching error.