I understand that when Akshan steals Shady Spectacles to his opponent, he can copy another unit and when it dies, the gear still goes back to his opponent since the "I" in "until I leave the board" refers to the game object. What happens when a unit with Deathknell with Shady Spectacles copies a unit without Deathknell and dies?

No, the Deathknell will not trigger.

When Shady Spectacles makes the equipped unit "become a copy" of the chosen unit, it replaces all copyable traits — name, type, Might, rules text, and keywords (Rule 472.1.b.3, patch notes 2026-03-30). Since the chosen unit has no Deathknell, the equipped unit loses whatever Deathknell it may have originally had. Deathknell is a characteristic of the permanent (Rule 808.3), and once overwritten by the copy, it's simply not there to trigger when the unit dies.

You're correct on the control side too — when Akshan dies, the "until I leave the board" control-change effect stops, and Shady Spectacles reverts to its original owner. The gear detaches from Akshan (Rule 435) and will be Recalled to the original owner's base during the next cleanup as unattached gear at a battlefield (Rule 452.1).