What your friend describes was discarded during the Advanced Class Guide playtest (found here, more specifically on this developer post). Those hybrid classes would not only count as both parent classes, but would also lock you out of those classes for multiclassing purposes. For example, a Skald would count both as a Bard and as a Barbarian, and you couldn't multiclass Skald/Bard or Skald/Barbarian at all.
That rule was discarded in the final version of the hybrid classes
And here are the final rules for Parent Classes:
Parent Classes: Each one of the following classes lists two classes that it draws upon to form the basis of its theme. While a character can multiclass with these parent classes, this usually results in redundant abilities. Such abilities don’t stack unless specified. If a class feature allows the character to make a one-time choice (such as a bloodline), that choice must match similar choices made by the parent classes and vice-versa (such as selecting the same bloodline). The new classes presented here are all hybrids of two existing core or base classes.
Now, they are their own thing, you can multiclass just fine, and you do not count as being a member of the parent class unless you have a special class ability saying that you do. For example, the Brawler's Martial Training says:
a brawler counts her total brawler levels as both fighter levels and monk levels for the purpose of qualifying for feats. She also counts as both a fighter and a monk for feats and magic items that have different effects based on whether the character has levels in those classes (such as Stunning Fist and a monk’s robe).
The Arcanist lacks an ability like this, and thus does not count as wizard and/or sorcerer for feats, magic items, prestige classes, or anything that depends on you being a member of either of the two classes.
Regardless, arcane discoveries are not feats. But an arcanist may take the Extra Exploit feat to learn additional exploits.
Best Answer
Yes.
The text of the Arcanist's ability specifies (emphasis mine):
There is no caveat that limits the choices; it says any school a wizard could choose, the Arcanist can choose. Additionally, the Arcanist was published well after other materials that introduced most of these variant arcane school choices, so it is not a situation where the ability was written and the options it provides were expanded later; these were available choices when the Arcanist ability was written and the text specifies "any" rather than providing a list. Taken together, there's no rules-as-written reason that the Arcanist should not be able to choose these variant schools, nor does it appear that such a restriction was intended (quite the opposite).