The Roman coin bears the image of Caesar, we bear the Image of God, a teaching the Church has taken from the words in Genesis.
There are many characteristics we possess that give a glimpse of how we resemble God, though perhaps not any single characteristic is sufficient.
Created at birth in God's image, then sealed with God's imprint at Baptism and reinforced at Confirmation, there are many human characteristics which mirror God to others. As Aquinas and Augustine have noted, our human faculty for reasoning is surely one of them and not simply our ability to think, but our capacity to know we are thinking. Meditation, contemplation, reaching out to the heavens, into the atom and above the visible world to an unseen God - are God-given and God-like qualities.
Communities formed by love for family (especially the covenant of Matrimony), friendships, neighborhoods, church are some of the many dynamic relationships which can mirror the Divine Community of Persons in the Trinity.
Our capacity for compassion, to be with others in suffering, to use our spiritual imagination to empathize with others, to imagine how the world might be better without slavery or nuclear weapons and to pray and work toward it are co-creative acts of making the world a more peaceful place.
Not one characteristic makes us carbon-copy of God, but a complex multifaceted resemblance.