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'. The idea is that we're going to attempt a Proof by Contradiction: we'll start by assuming that this premise is true, and look at case studies to find obvious cases where contradictions, or other obvious breakages of game logic occur.
"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 have its effects occur at their 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?
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.
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 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 Interpretation 2, then yes, a spell duplicated by Wish was 'Cast'. There wouldn't be any way to resolve its effects otherwise. But under Interpretation 1, then it comes down to what your DM says.