This post is about the words catholic, orthodox, and evangelical and what they mean. The first question is whether only one branch (denomination) of Christianity can legitimately use any of these words. The answer is No; many churches can use them.
The Niceno-Constantinopolitan Creed includes the words “In one holy catholic and apostolic Church”. So any church that accepts this creed has some claim on the word catholic (as well as the word apostolic).
Catholic means universal so any church that identifies with the universal church (whether as part of it or the whole of it is another matter) has a claim on this word. This includes every branch of Christianity, though the Church of Rome has taken it their moniker.
It is similar with the word orthodox. Any church that considers its doctrine to be orthodox Christianity has a claim on this word. That covers every branch of Christianity, although the churches of Eastern Christianity have taken it as their moniker.
The word evangelical simply means “pertaining to the gospel.” Any church which promotes the gospel has a claim on this word. That covers every branch of Christianity, though some Protestant churches (especially revivalist ones) have taken it as their moniker.
We could say that the words catholic, orthodox, and evangelical have generic and specialized meanings. Their specialized uses are usually capitalized. No one branch of Christianity has a monopoly on any of these words, though it sometimes seems so.