So when you unlock an armor ornament for a current season pass, it unlocks for all classes. All you need to do is get on a character of that class to claim it. However, if you purchase an armor ornament from the Monument to Seasons past with chronologs, it only unlocks on the class you purchased it on. You would need to buy it again on a different character to get that piece for a different class.
Is this discrepancy intentional? If it is not intentional, then what happens to people who bought a single class's version of an armor ornament, or to people who bought the same armor slot of multiple classes?
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Probably intentional. If you want it for your other characters you'll have to buy them. In-game. If you had reached the rank required to get the ornaments in the past. Then you'll need to access it outside of the game/bungie site.