If we look at this passage, 1 Tim 4:3, from a modern, American perspective, it does seem to point uniquely to the Catholic Church because we are one of a handful of institutions that practices both fasting and celibacy. But we also must note that the passage does not describe the practices in the Church today - where the Apostle warns against those who forbid marriage, no one in the Church is forbidden to marry; indeed, dozens of weddings will be celebrated this summer at my parish! And when the Apostle cautions against abstaining from meats, every Catholic is allowed to eat meat 359 days each year - when we abstain we do not do so to disparage the good that the Father provides us, but out of discipline in accordance with a fast.
However, if we lens the passage to Paul’s initial audience, primarily Timothy but also his mission church, who were 1st century Jews and Greeks in Ephesus, we have to widen our perspective to include groups other than the Church because both celibacy and fasting were commonly practiced by Jews, Christians, Gnostics and others. We already know (Acts 15) that some Jews within the early Church caused trouble by demanding adherence to the Mosaic Law, and they were not the only group to attempt to demand more, or preach a different Gospel, than what the Apostles taught.
Paul’s caution in 1 Tim 4:3 was not against fasting itself, since fasting is Biblically-warranted, for example: Joel 2:12, Mt 6:16-18, Ez 8:23, and notably - since it describes the early Church - Ac 13:2. We do not always have a description of how the fasting took place, but it is not unrealistic to think that abstaining from meat during particular fasts was part of it.
Likewise, celibacy was taught as a virtue and as a calling within the early Church. Hence Paul commands (in 1 Cor 7) that men are better to not marry, that widows should remain unmarried, but that those who cannot refrain from lusts should marry to avoid sin. We cannot say that Scripture supports a reading of 1 Tim 4:3 that would forbid a Church from preaching celibacy; rather we are to be cautious of a Church that would forbid marriage, which goes beyond preaching the virtue of celibacy to binding it upon all believers.
So who would fit the proscriptions in 1 Tim 4:3? Sects known as Gnostics, or more radical aesthetes who not only shunned worldly pleasures such as wine, meat, and marriage, but taught others to find salvation in the same path. Irenaeus, writing in the late 2nd century, names one such sect, the Encratites, but these weren’t the only ones who practiced such, either before or since. The various Cathari heretical groups, including Albigensians and Bogomils, who were active in Europe between the 12th and 14th centuries, sought spiritual perfection through complete avoidance of anything of a sexual nature, including marriage and eating animals (themselves the products of sexual union). Note that even today such groups exist - notably the Heaven’s Gate cult, some of whose members underwent castration in order to avoid any impure desires altogether, while many cults embrace vegan or vegetarian lifestyles as a matter of theological mandate.
So we should take Paul’s command in 1 Tim 4:3 to caution against a doctrinal mandate towards excessive asceticism, for such is not necessary to be saved and it is false to require it for salvation. We should not take it to forbid a practice or discipline that is either voluntary or limited, especially in light of Scriptural support for both voluntary celibacy and for fasting of various forms.