Who CAN identify as a Catholic? Anyone can. It may be a misrepresentation of the truth, but anyone can “identify” themself as a Catholic.
Who can identify as a member of the Church? This question is more ambiguous than you think. First off there are two different fundamentally different understandings of what the term “member of the Catholic Church” means. For most people in most situations the term “member of the Catholic Church” simply means that the person has been validly baptized and, if old enough, has made their first Holy Communion and their Confirmation.
The other meaning of “member of the Catholic Church” is more theological in nature. The Church is the mystical body of Christ and ANYONE who is a part of that mystical body is a member of the Church. First and foremost are the Saints who are recognized as such, and everyone who is in Heaven. This group of Church members is called the Church Triumphant. In addition, all who have died and been judged worthy of inheriting eternal life(whether they were identified as Catholic on earth or not) are also members of the Catholic Church (even though we are unable to recognize who, exactly, these folks are.) This group is known as the Church Suffering (ie they are in Purgatory). Lastly, there are all those living who, by grace through saving faith in Christ, are in a state of grace and who, if they died today would be judged worthy of inheriting Heaven. They are also members of the Catholic Church, whether they identify themselves as Catholic or not. This group is referred to as the Church Militant. It is entirely this group that you likely meant to discuss in your original question though it is unlikely you had any idea that the term Catholic Church has a common (visible members) meaning and a different, theological one.
We simply don’t know in any absolute sense who the living members of the Church are in the theological sense, but in the common sense it is largely as I mentioned above - baptized and confirmed by a Bishop in the Catholic Church commonly means you are Catholic.
Certain sins place you outside of communion with the Church if they are committed with full knowledge of their gravity and sinfulness and entered into with full consent of the will. Abortion is one of them, but any mortal sin (10 Commandments) causes you to fall from grace and be outside of the Church Militant. In such a situation the individual needs to be reunited with the Church. The normative means of accomplishing this is via Sacramental Confession and Reconciliation.
In short, it’s a bigger question than you realized when you asked it!
Blessings!