How moral reasoning is measured? According to this answer, there are some kind of points used for moral reasoning. Therefore, I'm interested now, how the amount of points given for reasoning? Is there some kind of ideal moral theory used for it, or is it just how consistent the reasoning is or what?
It really depends on your theory.  If you can write measurement of morality in google scholar you would find many morality theories and attempts to measure it such as (Ziv, 1976). Most theories, as far as I know, use stories and evaluate children's responces to those theories. I'm not aware of any scale to measure moral development.

By the way, one of the most commonly used theory of moral development is Kohlberg's stage based moral development theory (Kohlberg, 1971). In that article, although I have not read fully, Kohlberg talks about some stories he used to measure moral development.

Of course there are many critiques to this stage based measurement of moral development. (Bandura 2014) can guide you on current theories. I haven't read any of those articles but they seem to be good starting point.

References:

Bandura, A. (2014). Social cognitive theory of moral thought and action. In Handbook of moral behavior and development (pp. 69-128). Psychology Press.

Kohlberg, L. (1971). Stages of moral development. Moral education, 1, 23-92.

Ziv, A. (1976). Measuring aspects of morality. Journal of Moral Education, 5(2), 189-201.