The simulacrum spell has a limitation, to try to prevent a caster from creating multiple duplicates:

If you cast this spell again, any duplicate you created with this spell is instantly destroyed.

However, the wish spell says:

The basic use of this spell is to duplicate any other spell of 8th level or lower. You don't need to meet any requirements in that spell, including costly components. The spell simply takes effect.

It sounds like using wish to duplicate the effects of simulacrum would not trigger the limitation in simulacrum, because you're casting wish and getting the effect of Simulacrum but not actually casting simulacrum. Is this right?

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:

Bestow Curse

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".

Spirit Guardians

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).

Existing duplicates being destroyed is part of the effect of simulacrum. Using wish to duplicate simulacrum would also duplicate that effect of simulacrum.

Avoiding that effect of the duplicated spell would instead be using wish to do something beyond duplicating an existing spell.

Specifically, it would be casting wish to get the effect “casting simulacrum but without destroying my existing simulacrum”. Casting wish for that effect would be using it “to produce any effect other than duplicating another spell”, and would carry the usual stress risk of that use.

As Written

The rules as they are written are unclear about this effect. A literal reading might allow it, but reading the "spirit" of the two rules seems to suggest the first one would be destroyed. So, could go either way.

As Intended

The rules as the designers intend seems to be that no second duplicate is allowed, based on this tweet by rules designer Jeremy Crawford:

Take a look at the spell's last sentence.

But that doesn't deal with the wish edge case, so this is also inconclusive until we see a ruling on this edge case.

Final Call

The final arbitrator of any D&D game is the DM. That puts this one, I think, in the position of "You can keep them both if you can convince the DM."

Personally

It's wish... Which is so much DM territory already. I'd likely allow it, but I'd be tempted to give to give it some negative effects that increase the number of times the person uses the spell to get two at a time -- maybe a level of exhaustion for each time the player tries it, applied to the new duplicate if there is already one.

A simulacrum you wish for does destroy any duplicates

As you read, the trigger for destroying other simulacrum duplicates is the casting of simulacrum (emphasis mine):

If you cast this spell again, any duplicate you created with this spell is instantly destroyed.

Even though using wish simply duplicates the effect of simulacrum, you are still considered to be casting the spell. This is revealed indirectly based on an official ruling on the use of Twinned Spell (which also requires casting a spell) in the Sage Advice Compendium:

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.

This is further explored in an answer to another question by Abe Karplus. Since you are considered to be casting the spell, it shares the same effect as if you cast it normally, including destroying any previous simulacra.

I feel you may be misunderstanding the description of wish.

The basic use of this spell is to duplicate any other spell of 8th level or lower. You don't need to meet any requirements in that spell, including costly components. The spell simply takes effect.

This doesn't mean it's duplicating the effect of another spell you've cast it just means you're casting wish and getting the effects of simulacrum or some other spell. The purpose of it is to basically use wish as an emergency spell to be another spell that you need. Now you could make it duplicate the specific effects of a previously cast spell, but that's not the same thing and would fall under the later description.

You might be able to achieve something beyond the scope of the above examples. State your wish to the GM as precisely as possible. The GM has great latitude in ruling what occurs in such an instance; the greater the wish, the greater the likelihood that something goes wrong. This spell might simply fail, the effect you desire might only be partly achieved, or you might suffer some unforeseen consequence as a result of how you worded the wish.

Rules-as-Written vs Rules-as-Interpreted

Now whether or not you can use it to get a second simulacrum of the same creature is going to depend on if we go as written or as interpreted. If you read it as written you aren't casting simulacrum you're casting wish. Now if you choose to interpret it this way you can then create an endless number of them. For example you don't need to use the wish spell but rather have your copy do it and target you. Then have the next copy do the same. As long as you have the resources to do this you can make an endless number of simulacrums. Sure it may be time consuming, but it's still not balanced or in the spirit of the game. If there is an actual 5e official ruling on this use of wish, I was unable to locate it.

Sansuri's Simulacrum

I did however locate another question that mentions a spell in an official 5e published adventure, Storm King's Thunder. It's a more expensive version that allows for as many simulacrums of the same creature without the destruction of the previous duplicate.

ISBN-10: 9780786966004

ISBN-13: 978-0786966004