Premise
This question was 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?
Best Answer
Yes, it does.
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:
The Twinned Spell metamagic option reads:
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.