I will actually argue that it is very useful and wish that more people started to be more specific about their faith. Firstly, stating that one is Catholic/Lutheran/... demonstrates that one understand that one is not representing the entire Christianity. Secondly, it is useful for discussion as it makes one's dogmas/axioms more explicit. And thirdly, it allows better granularity as some general teachings are really Christian (like the resurrection and most of the Credo) while other are specific to the tradition one follows (like the sacraments).

For example, a Catholic would hesitate to receive a "Christian eucharist" as the Catholic and protestant understanding of the Eucharist is so profoundly different

A final point, given the subject of the thread, is that since the Catholic Church regards itself as founded by Christ, it is older than the Bible, and that the Bible was primarily written by the Church, for the Church, to complement Tradition. They would consider splintering off Tradition and leaving Catholicism as moving away from Christ (as Christ is head of the Catholic Church)