This is the basis of all religions. A Catholic man and woman will not give birth to a muslim Child. The Child will be indoctrinated to be Catholic. Had the Child been adopted to a Protestant couple, the Child would be Protestant. The majority are born into a religion (or the lack of thereof) or are heavily influenced by the surroundings (which is why religions are concentrated geographically).

Religion is rarely a choice, so yes, most Catholics living today and in the past are involuntarily Catholic. Whether or not they "like" to be Catholic or not, it is rarely a choice made through rational thinking and the possibility to "browse" and see what is the most believable option.

About "involuntary Catholics"

While that distinction isn't recognized by the Church, it's an understandable condition to consider. It all starts with baptism, as with the previous answer that you linked to.

Baptism provides a permanent mark on the soul

Semel catholicus, semper catholicus (per Canon Law)

(Canon 11) The law recognizes as Catholic anyone baptized in the Catholic Church or received into it¹.

If you are baptized Catholic, but naught else is done to raise you in the faith, the Church will still consider you Catholic (per Canon 11¹, among other considerations). This is directly related to the indelible spiritual mark on your soul that occurs during the baptism.

CCC 1272 Incorporated into Christ by Baptism, the person baptized is configured to Christ. Baptism seals the Christian with the indelible spiritual mark (character) of his belonging to Christ. No sin can erase this mark, even if sin prevents Baptism from bearing the fruits of salvation. 83 Given once for all, Baptism cannot be repeated.

It is important to recall that the Catholic Faith is not a single event; the intent of being brought into the Catholic Faith Community (at whatever age) is to live our lives in the Catholic faith community (the church is the body of Christ) and to (as best we can) configure our lives to Christ.

Would they be considered Catholic, up to such point that the person intentionally renounced this?

Yes (and even if they renounce their baptized faith ;-) ) .

The Church will not renounce its connection to that person, though a person may leave the faith community by word, deed, or both.

1279 The fruit of Baptism, or baptismal grace, is a rich reality that includes forgiveness of original sin and all personal sins, birth into the new life by which man becomes an adoptive son of the Father, a member of Christ and a temple of the Holy Spirit. By this very fact the person baptized is incorporated into the Church, the Body of Christ, and made a sharer in the priesthood of Christ.

Your question appears to be referring to a Lapsed Catholic

Lapsed Catholics are all over the place. In the past few years I have (as part of the RCIA ministry) been of assistance to help some lapsed Catholics return to the Faith, to get confirmed, and to convalidate their marriages. (I am not a clergyman of any sort; lay person, volunteer, catechist).

A lapsed Catholic is a baptized Catholic who is non-practicing. Such a person may still identify as a Catholic and remains a Catholic according to canon law.

The canon law support to the answer may seen a little indirect, but it provides a clear idea on how the Church sees the relationship between the Baptized and the Church.

Can. 11 Merely ecclesiastical laws bind those who have been baptized in the Catholic Church or received into it, possess the efficient use of reason, and, unless the law expressly provides otherwise, have completed seven years of age. {standard age of reason}

While there are a variety of reasons that people profess to leave or ignore the faith they were baptized into, the Church has an outreach program called Catholic Come Home. A core message is:

Regardless of why you left or got out of the habit of going to Mass, you can always come home and return to the practice of the sacraments and the fullness of relationship with Jesus Christ and the Church he founded. We are Catholic. Welcome home.

¹ New Commentary on the Code of Canon Law, ed. by John P. Beal, James A. Coriden, Thomas J. Green, Paulist Press, 2000, p. 63 (commentary on canon 11

Even the form of censure known as excommunication does not in itself make a person an ex-Catholic: excommunicated persons are "cut off from the Church", barred from receiving the Eucharist and from taking an active part in the liturgy (reading, serving at the altar, etc.), but they remain Catholics.[11] They are urged to retain a relationship with the Church, as the goal is to encourage them to repent {via the sacrament of penance and reconciliation} and return to active participation in its life.

Those who baptize infants are bound by Canon Law to raise those children Catholic. A Catholic parent must

Can. 1125 1/…do all in his or her power so that all offspring are baptized and brought up in the Catholic Church.

Thus, unless the Catholic parents utterly fail in their duty, it is nearly impossible for a child to be baptized Catholic and not realize once the child reaches the age of reason.

Secretly baptizing foster children, for example, is prohibited by the Church for the reason that it cannot be assured that the child will be brought up in the Catholic religion.

Can. 868 §1. For an infant to be baptized licitly [i.e., lawfully]…2/ there must be a founded hope that the infant will be brought up in the Catholic religion; if such hope is altogether lacking, the baptism is to be delayed according to the prescripts of particular law after the parents have been advised about the reason.

Thus, to answer your questions:

a Catholic baptism alone makes a person a Catholic

This isn't quite correct. Catholic baptism + raising the infant Catholic makes the infant Catholic. If the infant is baptized Catholic but not raised Catholic, how is this any different from if the infant were baptized outside the Catholic Church? Anyone, inside or outside the Church, who can apply water and say the baptismal formula can validly baptize.

Catholic teaching is not that baptism alone makes one a member of the Church. As Pope Pius XII's encyclical Mystici Corporis Christi said,

22. Actually only those are to be included as members of the Church who have been baptized and profess the true faith, and who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed.

Baptism and profession of the Catholic Faith are both required to be members of the Church. Since infants lack the use of reason, the Catholic environment in which the infant is brought up substitutes for the inability of the infant to profess the Faith. After the infant grows up and attains the age of reason, he can formally make this act of faith in the sacrament of Confirmation.