What are the main features in a Catholic understanding of the human person? What elements of this understanding have parallels in Greek philosophical thinking?
The question is rather broad.
The Greeks and Ancient Hebrews were in contact with each other,
but didn’t have a particulalrly strong mutual influence. Christian philosophy drew from both Hebrew and Greek/Roman sources.
To answer the question properly you would need to be familiar with totally non-Western ideas. Then you you pick out the common elements in the Greek/Hebrew/Christian understanding. That is beyond my knowledge.
However I’d say that both Greeks and Christians believe that man is not of the Gods, but has the capacity to become God-like. Both believe that humans have will which they can oppose to God or the gods. Neither believe that man is essentially the same as the animals.
Thus differs from, say, ancestor worship, where deceased humans are gods. It differs from fatalism, where destiny is all mapped out by the stars. It also differs from animism, where the beasts are considered to have human-like souls.
However your question is rather deep. It needs a book to answer, not a post.