Tags have no rules of their own.
From the introduction section of the Introduction to the Monster Manual (or the corresponding part of the Basic Rules):
The parenthetical tags provide additional categorization for certain creatures. The tags have no rules of their own, but something in the game, such as a magic item, might refer to them.
Therefore, tags have no effect on the creature's Type. The Elf tag does not indicate humanoid or fey, it is merely a category marker.
So, no, hold person does not affect Eladrin since they do not have the Type of Humanoid.