Being harnessed isn't copiable. If something else becomes a copy of The Soul Stone, it must be harnessed separately. Similarly, if The Soul Stone is already harnessed and becomes a copy of something else, it stays harnessed, though it's very likely that won't be relevant until that copy effect ends.

Until it is harnessed, The Soul Stone doesn't have the ability listed after the infinity symbol. It also doesn't have that ability in zones other than the battlefield.

"Harnessed" is a designation that the permanent has once an ability instructs you to harness it. It has no special rules meaning other than being a designation that the ∞ ability (and, theoretically, other effects) can see.