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The detachment rule for the Plague legion says:

While an enemy unit is within 9" of one
or more Legiones Daemonica Nurgle
units from your army, that enemy unit
is within your army’s Shadow of Chaos.
In each player’s Command phase, select
one enemy unit within your army’s
Shadow of Chaos. That unit must take a
Battle‑shock test.

My question is that in my opponents command phase does he select one of his own units, as in an enemy unit from my perspective, or does he select one of my units, as in an enemy unit from his perspective?

I think it is telling you, the person who brought the detachment with the ability, to do the selecting each time.

It definitely says that you, the Nurgle player, to select one of your opponents units, in both your and their Command Phase, to take a Battle Shock test.