Adoption is a legal set of rights and obligations. Let's pick to random people. To determine whether adoption laws discriminate, the question is "Can person A adopt person B?" With my proposed law, controlling for all other requirements and qualifications, the answer would be "Yes" if person A has not already adopted another person and "No" if person A has. Based upon this analysis, we can determine that adoption laws treat persons differently based upon whether they have previously adopted someone. That is discrimination.
People are far too afraid to admit the existence of discrimination. Discrimination is not inherently evil and it is not, by itself, make laws unconstitutional. As you noted, adoption laws already permit persons to take into consideration their prior adoption history. Since the status of being an adoptive parent is not protected, the government only needs a rational basis to engage in this type of discrimination. It is fairly obvious that one's adoption history is a relevant factor in determining one's ability to fulfill the obligations of an adoptive parent.