A lot of non-denominational Christians think Catholics are "not Christians" because they call themselves "Catholic".
In reality, the Catholic Church was the first Church, and the name Catholic comes from one of St Paul's letters, from the Greek "ekklesia kata holos", "the universal church".
The name "Christian" was given to the Catholics in Antioch (where Peter was Bishop before moving to Rome).
St. Pacian ( 310–391 AD ) explained this way: "Be not troubled: Christian is my name, Catholic my surname. The one designates me, while the other makes me specific. Thus am I attested and set apart.".