My understanding is that, in the US, a bishop may request that a person or organization in his diocese not publicly represent themselves as Catholic. However, there is no legal obligation for them to comply. Their willingness to comply is based upon their respect for the office of the bishop.
There is a legal obligation to comply, that is the very real canonical obligation of Canon Law. Unfortunately, many people regard Canon Law as a lesser code and only worthy of respect, rather than obedience; the penalties for violating Canon Law are fewer and far between, don’t involve jail or fines, and an organization that dissents from the Church is unlikely to care what the Church thinks is binding, even if that organization is composed of baptized Catholics who are subject to the law.
The ordinary can go a step further and issue individual
latae sententiae excommunications for those who choose to be members of these groups. +Bruskewitz of Lincoln did exactly that for many groups, a while ago. Once again, if you’re a dissenting Catholic, these excommunications don’t have much teeth, and the ability and uniformity of enforcement, e.g. priests denying Holy Communion, is uneven.
We have seen at least one recent example of compliance with the canonical request of a bishop, that is RealCatholicTV changing their name to
ChurchMilitant.TV.
The word “Catholic” is not trademarked. Many organizations and religious denominations not in communion with Rome, and often with sharply different beliefs than presented by the Holy See, are legally free to use the word “Catholic.” At least that is the situation in the US.
That is the view from civil law. Canonically, the view is about the same all over the world, as I have explained above.
