Premise

Inspired by the line of reasoning in this answer. There are a number of spells that have wording "when you cast" or "as you cast" included. Specifically, some effects have predicates like "when you cast" in order to come into effect. If wish does not count as casting the a duplicated spell, it could have ramifications of the resulting effects.

Examples

Call lightning requires choosing a location

When you cast the spell, choose a point you can see under the cloud. A bolt of lightning flashes down from the cloud to that point. Each creature within 5 feet of that point...

Teleportation circle requires drawing a circle as you cast the spell for the portal to show up in.

As you cast the spell, you draw a 10-foot-diameter circle on the ground ...

Does a wizard duplicating a spell with wish count as casting the duplicated spell?

Yes.

This is never explicitly stated, but we can infer it from the Sage Advice Compendium which notes that the sorcerer's Twinned Spell feature works on spells duplicated by wish:

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

Twinned Spell reads

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).

The wish spell has a range of self, so Twinned Spell must be applying to the casting of the duplicated spell.

Also, just as a matter of common sense, there's no reason wish would be able to duplicate all but a rather arbitrary subset of spells.

A lot of weird things happen if a Duplicated Spell is not 'Cast'

In this post, I am not necessarily going to be making an argument as to whether a spell cast by Wish counts as being 'Cast' or not (well, maybe at the end...). I am, however, going to make arguments about what the consequences would be if using Wish to duplicate a spell means the spell in question has not been 'Cast'.

All of the following sections involve considerations that become true if a spell duplicated by Wish is not, mechanically, 'Casted'.

"When this spell is cast again..."

As mentioned in the linked question about Simulacrum, any spell that has a clause reading something like "if this spell is cast again..." would not have that clause take place. Simulacrums will stick around, you can summon multiple Spiritual Weapons, and so on.

"If you cast this spell with a higher spell slot..."

Any spell duplicated by Wish would be cast at its lowest level, and receive no benefits from being up-cast at a higher level of spell slot.

"As part of the action used to cast this spell..."

If the spell says that something does happens "as part of the casting of the spell", then that something doesn't happen.

This means that spells like Booming Blade and Green Flame Blade have no effect: they both stipulate that something occurs during the casting of the spell:

As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell's range, otherwise the spell fails. [...]

—Booming Blade and Green Flame Blade, Sword Coast Adventurer's Guide, pg. 142 & 143

This also affects spells like Teleportation Circle ("As you cast the spell, you draw a 10-foot-diameter circle on the ground ...") or Magic Circle ("When you cast this spell, you can elect to cause its magic to operate in the reverse direction...").

Esoteric: can a spell's casting be "wasted" if it was never cast?

You and up to eight willing creatures within range project your astral bodies into the Astral Plane (the spell fails and the casting is wasted if you are already on that plane).

—Astral Projection, Player's Handbook, pg. 215

This is where the RAW becomes unclear: what happens if a spell fails without having been 'Cast'? What does "the casting is wasted" mean for a spell that wasn't cast?

"The effects of the same spell cast multiple times don't combine..."

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.

—Combining Magical Effects, Player's Handbook, pg. 205

If the second spell produced by Wish was not 'Cast' into being, then this rule would not apply. After all, the rule specifically says that the spell has to be 'Cast' multiple times, and this would be the case from nearly any other source, including an Innate Spellcaster, but not from spells duplicated by Wish.

You can't Target yourself with an AOE spell

If a spell targets a creature of your choice, you can choose yourself, unless the creature must be hostile or specifically a creature other than you. If you are in the area of effect of a spell you cast, you can target yourself.

—Targeting yourself, Player's Handbook, pg. 204

So if you're duplicating a spell that normally allows you to choose targets in an area of effect, you'd no longer be able to target yourself.

Let's try another angle...

From all that I've listed above, it's clear that the mechanical consequences of spells duplicated by Wish, if they don't count as being 'Cast', are quite significant. But the rules being weird is not sufficient proof that they're wrong or misinterpreted.

What does the chapter's opening paragraph say about Spellcasting?

Spellcasting

Magic permeates the worlds of D&D and most often appears in the form of a spell.

This chapter provides the rules for casting spells. Different character classes have distinctive ways of learning and preparing their spells, and monsters use spells in unique ways. Regardless of its source, a spell follows the rules here.

—Spellcasting, Player's Handbook, pg. 201

So here's an interesting passage: in the same paragraph, the book says both "this chapter provides the rules for casting spells", but also says "regardless of its source, a spell follows the rules here".

There's [at least] two ways to interpret that.

One way is to say that that last sentence is an exception to the former: the chapter contains rules for casting spells, but spells produced by "some other means" still also obey those rules.

The other way is to say that the last sentence is implying that any spell produced by any source is obeying the same rules here because, implicitly, a spell produced by any other means has been 'Cast'.

So in one way, it might come down to how you interpret this paragraph. Under the latter interpretation, then yes, a spell duplicated by Wish was 'Cast'. There wouldn't be any way to resolve its effects otherwise.

Under the former interpretation... Well, unless someone finds the smoking gun I missed, then it comes down to what your DM says.