A spell's rule text describes what happens when you cast the spell. What happens when you cast it IS "the spell's effect". A spell "taking effect" and a spell "being cast" are one and the same thing.
The phase "The spell simply takes effect" is contrasting with whatever normal procedure for casting the spells is (in terms of components, etc), it is not saying "the effect of the spell happens but it does not count as having been cast".
If we did apply a hyper-literal reading that "when you cast this spell again" only refers to actually casting Simulacrum by the normal process, not using any other way of getting the effect, then we would be forced to include that any spell clause containing the language "when you cast this spell" doesn't actually take effect when you produce its effect using Wish.
A couple of random examples I pulled up on roll20.net:
When you cast this spell, choose the nature of the curse from the following options
Nope, when you wish for Bestow Curse, nothing happens, because you didn't "cast this spell".
When you cast this spell, you can designate any number of creatures you can see to be unaffected by it.
Wished for Spirit Guardians harm your friends, nothing you can do about it.
And as David Coffron points out in his answer, a Sage Advice Compendium has given us:
Can my sorcerer use Twinned Spell on a spell duplicated by the casting of a wish spell? And if so, how many sorcery points does it cost? Yes, you can. It costs the number of sorcery points appropriate for the level of the spell you’re duplicating.
The text of Twinned Spell:
When you cast a spell that targets only one creature and doesn’t have a range of self, you can spend a number of sorcery points equal to the spell’s level to target a second creature in range with the same spell (1 sorcery point if the spell is a cantrip).
To be eligible, a spell must be incapable of targeting more than one creature at the spell’s current level. For example, magic missile and scorching ray aren’t eligible, but Ray of Frost is.
This ruling wouldn't make sense if casting Wish to so that "the spell simply takes effect" did not count as you casting the duplicated spell.
It is quite clear that the text "when you cast this spell" in a spell's description is just referring to the event of you invoking the spell's effect (specifying that you must make the choice of curse or unaffected creatures in the examples above at the moment the spell's effect begins, not at some other time), not making a distinction between "casting" and any other ways of making the spell take effect. Otherwise wish (and any other rules that refer to the effect of a spell without being normally cast) would produce strangely different behaviour for many spells (making some of them do nothing at all).