There is a quote by St. Augustine that says “And so, lastly, does the very name of 'Catholic', which, not without reason, amid so many heresies, the Church has thus retained; so that, though all heretics wish to be called Catholics, yet when a stranger asks where the Catholic Church meets, no heretic will venture to point to his own chapel or house.”
Except, it seems that there are some churches which have indeed the audacity to claim for themselves the name "Catholic", when they have actually embraced not-a-few heresies*.
Another example is the American National Catholic Church, which has parishes in Connecticut, Florida, New Mexico and New Jersey.
And then there's the Old Catholic Church, the Independent Catholic Church, the Brazilian Catholic Apostolic Church, the Catholic Apostolic Church of Antioch at Santa Fe, and a whole slew of others.
*NB: While I do assert that these churches have embraced heresies, I am not being so bold as to call these folks heretics. That's above my paygrade.I leave that to the Church to declare someone to be a heretic.
These are churches which have usurped the name of the Catholic Church, and many unsuspecting Catholics have participated in liturgies here unwittingly.
These "independent" Catholic churches cheerfully profess that they're not subject to the Roman pontiff. They have said, "Thanks, but no thanks, we're fine on our own without a pope and it's perfectly acceptable to do that and still be Catholic!"
However, as Catholic apologist Dave Armstrong states: it's absurd to think of a group of Catholics being subject to the Pope, and then another subgroup of "Catholics" who are not subject to the Roman pontiff.
If you want to claim the name "Catholic" for your church, you ought to have some truth-in-advertising and profess from the pulpit all that the One Apostolic Holy Catholic Church professes. The decent thing for these churches to do would be to change their name to: Episcopalian. Or Protestant. Or anything other Christian denomination which has its roots in Catholicism, and then divorced itself from the Root.
These "Catholic" churches have taken all the hard teachings of Christ, re-conformed them to something more palatable, something more in conformity with their own personal views ("I like it when everyone gets to go to communion, even if you're not in communion with my church!" "I don't like it when women can't be priests!" "God wouldn't do [fill in the blank], and it's just coincidence that I don't like it when we have to do [fill in the blank]")
Yet, as Pastor Tim Keller states, "If your god never disagrees with you, you may be worshipping an idealized version of yourself".
We all should be part of the One Holy Catholic and Apostolic Church, despite our own views on particular issues, without leaving to find a church that agrees with our sensibilities. And, like a math student who struggles to get to the correct answer her professor gives, re-configure, adjust and conform our views so it agrees with the Professor.
So, if you were on vacation and went to Mass at a local Catholic church but found something wonky there, it most likely wasn't an actual Catholic church.
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However! Just to complicate things a wee bit...
There are also some Catholic Churches, fully in union with the Holy Father, which are legitimately Catholic, even if they're not ROMAN Catholic.
The Catholic Church consists of several major rites (of which the Roman, or the Latin rite, is the largest), and a multitude of smaller "churches".
All of us can legitimately attend Mass at any of these parishes. There may be some variance in customs or traditions (small "t" tradition, not Sacred Tradition), but their doctrines do not deviate from the faith, given once for all, to the Church.
And if they do something different than we normally see at our Roman Catholic liturgies, it is not a wonky thing, but rather just another way to give praise and worship to the Lord.
