The concept of "identity moratorium" comes from the identity status theory of psychologist James Marcia. This theory is concerned primarily with adolescence and extends into young adulthood. "Moratorium" is one of four statuses. Another, which you mention in your question, is "achievement." Both of these are statuses, or states of being at the end of a process. Your question suggests that you are instead approaching them as events or as "light bulb" moments of realization, which is different from the way Marcia presents them.
Marcia’s idea is that individuals start constructing personal identities in childhood. These identities develop further while they are adolescents and into early adult years. At different times and stages, an individual can occupy one of four statuses, including the two about which you ask.
A key element of this body of theory is that development is strongly social in character. Each person is influenced by their interactions. While achievement status can be reached by younger persons, it is mostly associated with young adults. Achievement here involves commitment to moral values and to personal goals. Individuals may learn new information and explore new beliefs, but those things will not really shake the values they have already established.
People in the moratorium category, in contrast, have not yet cemented their identities and life goals. They are likely to do more exploring and may be swayed by the new experiences they encounter. This identity-seeking, including more concern with short-term problems, is associated with—and considered normal for—adolescents.
In this story, Bruno is put into a situation where his opportunity for normal exploration is abruptly cut off. He is not only separated from his social peers at home, but when he arrives at the camp, there are no potential peers. His lack of understanding, as his parents try to keep information from him, is intensified by his loneliness. Meeting and befriending Schmuel, which is a new experience, is an aspect of the moratorium status: so is his exposure to, and rejection of, numerous elements of Nazism.
The figurative idea of "trying on new identities" is here presented as literal, when Bruno tries on the "new" (as in different) clothes that Schmuel wears: the cap and pajamas.
Because Bruno's parents withhold information, Bruno can't know the danger he has placed himself in through this experimenting. And because this combination of factors ends his life, he is forever blocked from moving into the achievement status. He can never become an adult.