I think it was fairly common in his day, but is it long out of favor? Would it be an anachronism to name a child Plato today?
It would be Platon in Greek, and the Wiki diambiguation page for that name has no Greek with first name Platon and an Wikipedia article past the 8th century AD (the others are Orthodox non-Greeks and one Greek born in London)
I believe the current Greek form of the name is “Platonas”. It’s also used as a surname but there seem to be a fair few Greeks who have it as a first name, such as the football player Platonas Karakatsanis, the actor Platonas Brakoulias, the government official Platonas Zafeiris, etc.
The name day of Platonas in the Orthodox calendar is November 18, fyi.
Keep in mind Plato was a nickname. His real name was Aristocles.