...by those who don't know anything about Judaism or Hebrew.
I'd like to see who "generally acknowledges" this nonsense. The Bible itself makes clear that Coniah is a shortened version of the name Jeconiah and he is the same person (First Chron 3:16-17, Jeremiah 24:1, 27:20, etc.). He is the son of King Jehoiahim, and apparently he was also known by a name awkwardly similar to his father's (Jeremiah 52:31) which may explain the frequent use of an alternate name to identify him.
To be a cohen, he'd have to be a patrilineal descendant of Aaron and it appears, that as King Johaiahim's son, he positively was not a cohen.
On the other hand, he was an ancestor of Jesus (Matthew 1:11-12), which, despite being defeated by the Babylonians, is something to brag about.