The Catholic Encyclopedia article titled "Unity (As a Mark of the Church)" says the following in a paragraph with the heading "Some False Notions of Unity":
All admit that unity of some kind is indispensable to the existence of any well-ordered society, civil, political, or religious. Many Christians, however, hold that the unity necessary for the true Church of Christ need be nothing more than a certain spiritual internal bond, or, if external, it need be only in a general way, inasmuch as all acknowledge the same God and reverence the same Christ. Thus most Protestants think that the only union necessary for the Church is that which comes from faith, hope, and love toward Christ; in worshipping the same God, obeying the same Lord, and in believing the same fundamental truths which are necessary for salvation. This they regard as a unity of doctrine, organization, and cult. A like spiritual unity is all the Greek schismatics require. So long as they profess a common faith, are governed by the same general law of God under a hierarchy, and participate in the same sacraments, they look upon the various churches -- Constantinople, Russian, Antiochene, etc. -- as enjoying the union of the one true Church; there is the common head, Christ, and the one Spirit, and that suffices. The Anglicans likewise teach that the one Church of Christ is made up of three branches: the Greek, the Roman, and the Anglican, each having a different legitimate hierarchy but all united by a common spiritual bond.
Likewise, Pope Pius XII, in his encyclical Mystici Corporis Christi writes:
Hence they err in a matter of divine truth, who imagine the Church to be invisible, intangible, a something merely "pneumatological" as they say, by which many Christian communities, though they differ from each other in their profession of faith, are untied by an invisible bond. (Mystici Corporis Christi, 14)
The Catholic Encylopedia article referenced above continues by describing the "True Notion of Unity":
The Catholic conception of the mark of unity, which must characterize the one Church founded by Christ, is far more exacting. Not only must the true Church be one by an internal and spiritual union, but this union must also be external and visible, consisting in and growing out of a unity of faith, worship, and government. Hence the Church which has Christ for its founder is not to be characterized by any merely accidental or internal spiritual union, but, over and above this, it must unite its members in unity of doctrine, expressed by external, public profession; in unity of worship, manifested chiefly in the reception of the same sacraments; and in unity of government, by which all its members are subject to and obey the same authority, which was instituted by Christ Himself. In regard to faith or doctrine it may be here objected that in none of the Christian sects is there strict unity, since all of the members are not at all times aware of the same truths to be believed. Some give assent to certain truths which others know nothing of. Here it is important to note the distinction between the habit and the object of faith. The habit or the subjective disposition of the believer, though specifically the same in all, differs numerically according to individuals, but the objective truth to which assent is given is one and the same for all. There may be as many habits of faith numerically distinct as there are different individuals possessing the habit, but it is not possible that there be a diversity in the objective truths of faith. The unity of faith is manifested by all the faithful professing their adhesion to one and the same object of faith. All admit that God, the Supreme Truth, is the primary author of their faith, and from their explicit willingness to submit to the same external authority to whom God has given the power to make known whatever has been revealed, their faith, even in truths explicitly unknown, is implicitly external. All are prepared to believe whatever God has revealed and the Church teaches. Similarly, accidental differences in ceremonial forms do not in the least interfere with essential unity of worship, which is to be regarded primarily and principally in the celebration of the same sacrifice and in the reception of the same sacraments. All are expressive of the one doctrine and subject to the same authority.
True unity is triadic; it consists in unity of faith (doctrine), unity of worship (cult), and unity of authority (government). These correspond to Christ's threefold role as Prophet, Priest, and King. Any form of unity that excludes one or more of these three does not merely reduce the degree of unity that marks the true Church, it destroys the possibility of true unity. That is because all three bonds of unity are essential for unity; each depends on the other two. (I discussed this in more detail under the heading "Three Modes of Organic Unity" in my post titled "The Sacrilege of Schism".) Those who say "We're all under the same authority, i.e. Jesus", and use that to deny that unity of ecclesial government requires that someone have episcopal primacy do not seem to realize that they have thereby undermined all ecclesial authority, and in principle conceded everything to the person who says, "I don't need to be under the authority of any pastor; I have Jesus." This is why there is no [consistent] position between individualism (which is necessarily disposed to fragmentation) and Catholicism.
Lord Jesus, may we all be truly one, so that the world may know that the Father sent You. (John 17:23)