To iterate , the eremitic life is of direct consequence and concern, obviously, to all Catholic Church hermits--all of whom have professed their vows which must include (or could simply state) the three Evangelical Counsels which are celibacy, poverty, and obedience.
The two previous posts cited the appropriate institutes and law as to licit and credible options for Catholic hermits whose adherence to the requirements of the Church, and for some, also CL603, are thus living in the state of the consecrated life of the Church. [See The Catechism of the Catholic Church, Section 3, Parts 914-921; and Can. 603, §1 and §2.]
Also note that the Church is comprised of three categories: Hierarchy, Laity, and Consecrated Life. [See The Catechism of the Catholic Church: Article 9, Paragraph 4. Christ's Faithful - Hierarchy, Laity, Consecrated Life, 871-873.] Of these, those in the Consecrated Life may derive from either the hierarchy or the laity.
Thus, those in Holy Orders, or Hierarchy, such as priests and bishops, may be called to consecrated life, as in priest hermits or priest religious brothers. In fact, there are and have been bishops, cardinals, and Popes in the Consecrated Life of the Church as hermits and/or religious brothers (monks). To do so, of course, requires for a bishop to no longer actively, publicly as recognized by Church law, fulfill his episcopal office. Even so, such priest or bishop hermits derive from the Hierarchy. And, those in the Consecrated Life of the Church (hermits, consecrated virgins, widows, religious brothers and sisters) may also derive from the Laity of the Church.
In practice, those of the Hierarchy (priests, bishops, cardinals or even a Pope) who enter/s the Consecrated Life of the Church are not referred to or do not to refer to themselves as hierarchy hermits, hierarchy brothers or hierarchy monks--even though they are derived from the Hierarchy. Neither are or do those of the Laity who enter the Consecrated Life of the Church referred to as or refer to themselves as laity hermit, laity virgin, laity widow, laity brother/monk, or laity sister/nun--even though they are derived from the Laity. (This is true for hermits whether or not their vows include the proviso of CL603.)
Thus, Catholic hermits who by their vows and life in accordance to the institutes of the Church have entered into the Consecrated Life of the Church are Church. By professing the required three evangelical counsels (celibacy, poverty, obedience) and fulfill the conditions of hermit life as delineated in The Catechism in pertinent section cited, they are no longer married or single laity. For example, marriage vows do not include celibacy and poverty; obedience is to be husband to wife and wife to husband.
Of course, all Catholic Christians (whether of the Hierarchy, Consecrated Life, or Laity) are to be obedient to God, to the Church, to the Word of God, to Hierarchy, to Superiors--all according to one's state in life. (As Christian children, we are to be obedience to our parents; as Jews and Christians, obedient to the Ten Commandments, etc.)
What is cited in The Catechism of the Catholic Church and in the proviso of CL603, and by virtue and fact of the specific vows required of each state of life in the Church, should suffice to explain why consecrated Catholic hermits (and also the consecrated virgins and widows) are part of the Consecrated Life of the Church--although they have derived from the Hierarchy or the Laity.
Likewise, consecrated Catholic hermits (virgins, widows, religious brothers and sisters) are not the Hierarchy of the Catholic Church, as in Holy Orders of priests and bishops.
The options for Catholics to enter into the Consecrated Life of the Church as a hermit are either for those in the Hierarchy or for those in the Laity to discern, receive, and accept a call from God to the eremitic life.
The Catholic priest or bishop, or lay person (including priest, brother, or lay person, who is currently a religious brother or sister in a Catholic Church-condoned religious order or community), may aspire to the consecrated state of life as an eremite. He or she then must fulfill the requirements in profession of vows and manifest the eremitic life in accordance with the cited specifics in The Catechism of the Catholic Church, of the institutes of the Consecrated Life of the Church. And, if the aspiring hermit requests and a bishop agrees, then he or she must fulfill the additional provisions in Canon Law 603.