Does a creature enchanted by an aura that gives it a triggered ability lose that ability if the aura's effect is somehow copied onto another creature?

An Aura's granted abilities are continuous effects that apply to the permanent the Aura is attached to (its enchanted object). This is governed by CR 611.3d, which states that a continuous effect from an Aura applies to the object the Aura is attached to. If the Aura is physically attached to Creature A, Creature A gets the triggered ability — full stop.

When someone 'copies' an Aura's effect onto another creature (for example, via a card like Strionic Resonator copying a triggered ability that originated from an Aura, or an effect that replicates what an Aura does), that is a separate effect acting on the second creature. It does not move the Aura enchantment itself, and thus does not change what the Aura is attached to. The original Aura remains on Creature A, and Creature A continues to have the granted ability (CR 611.3, 611.3d).

Auras can only be moved or transferred by specific game mechanics — such as effects that say 'attach' or 'enchant target,' or replacement effects that re-attach the Aura. Simply duplicating an Aura's static or triggered effect elsewhere does not unenchant or re-enchant anything (CR 701.3).

Concrete example: Creature A is enchanted by Curious Obsession, giving it 'Whenever this creature deals combat damage to a player, draw a card.' An opponent uses Copy Enchantment to create a token copy of Curious Obsession attached to Creature B. Now Creature B also has the triggered ability. But Creature A still has its original triggered ability from its own Aura — it did not lose anything. Both creatures independently trigger when they deal combat damage.

So the original enchanted creature never loses its Aura-granted ability just because another creature somehow gains the same or a copied effect. The ability is tied to the Aura's attachment, and that attachment did not change (CR 611.3d, 303.4).