No, man is not triune. “Triune” is a word combining “tri” and “unity.” To say that man is triune assumes, for one thing, a tripartite anthropology, which many do not accept. For another thing, even if one assumes a tripartite anthropology, man cannot be triune in any sense the way God is, since with God’s triunity come simplicity, consubstantiality, etc. Man might be three, but three parts, each of different substances.
Saying man is triune might sound clever, but it is confusing, at best.