To what extent (if any) does the Catholic Church recognize adoption? Assuming the Church recognizes adoption, if a man or woman were to invite another to live together as a loving relationship as parent and child, but without any secular legal process, would the Church recognize the adoption?

I ask this because some Catholics in history and today have lived/live in a land where there is no secular legal process to recognize an adoption.

If the legal document is required, why does secular law have such an impact on Church teaching? (In some countries adoption is completely illegal.)

By "recognize" I mean if it would be a mortal sin if at some point in time the 2 parties changed their minds on the nature of the relationship and decided to get married, or the parent giving up and cutting all ties with the child. So the adoption being just as permanent and having the same dignity as a biological parenthood.
…without any secular legal process, would the Church recognize the adoption?


No.

Canon Law defers to civil authorities for the legal recognition of adoption. For example, adopted children must be legally recognized as such before adoptive parents can baptize them (Can. 877 §3).

1983 Code canon 110 deals with adoption:


  Children who have been adopted according to the norm of civil law are considered the children of the person or persons who have adopted them.


Regarding


  why does secular law have such an impact on Church teaching?


see:


  Can. 22 Civil laws to which the law of the Church yields are to be observed in canon law with the same effects, insofar as they are not contrary to divine law and unless canon law provides otherwise.


It's possible that in the future the Church might devise a "mechanism or procedure…to secure the institute of adoption."

For more information on the divine, natural, civil, and ecclesiastical laws aspects of marriage, see How Marriage Became One of the Sacraments: The Sacramental Theology of Marriage From Its Medieval Origins to the Council of Trent by Philip L. Reynolds, §16.5 "Thomas Aquinas on Marriage in Law" pp. 686-715 (PDF pp. 718-747).