Because the clone will enter the battlefield as a Harmonic Sliver that has 2 instances of "When this permanent enters the battlefield, destroy target artifact or enchantment": one instance from its own ability, and one instance from the Harmonic Sliver that was copied. Multiple instances of triggered abilities are not redundant.
After it gains the abilities from itself and the other Harmonic, it now has these abilities:
When this permanent enters the battlefield, destroy target artifact or enchantment.
When this permanent enters the battlefield, destroy target artifact or enchantment.
Signature by Inkfox Aesthetics by Xen
[Modern] Allies
After it gains the abilities from itself and the other Harmonic, it now has these abilities:
When this permanent enters the battlefield, destroy target artifact or enchantment.
When this permanent enters the battlefield, destroy target artifact or enchantment.
Each instance of the ability triggers separately.
Signature by Inkfox Aesthetics by Xen
[Modern] Allies
2 Harmonic Slivers work the same way - right? It doesn't have to be a Clone for that to work.
Yes, sure. The entering Harmonic Sliver will have two instances of the triggered ability as it enters the battlefield.