Definition
The Greek word χαρακτηρ denotes "an engraving" (source).
Translation
The modern day translation "imprint" of ESV perfectly corresponds to this meaning.
The translation "exact likeness" is actually not a literal translation but a dynamic equivalence.
Exegesis
The Lord Jesus is a copy of God's being. It means that Jesus is not the original being but a mere copy of another's being.
However, copies contain everything the original contains. Therefore, the Lord Jesus owns everything God owns.
The ESV translation has faithfully preserved the meaning of the Greek in contemporary 21st century English:
"...exact imprint of his nature..." [χαρακτηρ της υποστασεως αυτου]
Q & A: In Hebrews 1:3 does χαρακτηρ της υποστασεως αυτου say that Jesus is the same substance or similar substance to God?
Based on the exegetical analysis we have, the interpretation that Jesus is of similar substance with the Father is open to Hebrews 1:3
Likewise, Jesus is of same substance with the Father is also a possible interpretation.
It seems that both the Unitarian and the Trinitarian can equally hold onto Hebrews 1:3 in supporting their differing Christological views. Therefore, it depends on one's tradition/faith to consider what interpretation is preferred.
