No, advanced monsters/NPCs with class levels in a single class are not multiclassed
As the DMG describes, you can add class levels to an existing creature to make it more powerful:
You can use the rules in chapter 3 of the Player’s Handbook to give class levels to a monster. For example, you can turn an ordinary werewolf into a werewolf with four levels of the barbarian class (such a monster would be expressed as “Werewolf, 4th-level barbarian”).
Start with the monster’s stat block. The monster gains all the class features for every class level you add, with the following exceptions [...]
The exceptions being that a monster doesn't automatically get starting equipment, it keeps the same hit die size it already has as a monster, and it uses its CR, rather than total levels, to calculate proficiency bonus.
Note that this refers to the rules in Chapter 3 of the PHB, which is the section on classes. The multiclassing rules are optional rules and actually feature in Chapter 6 - and these rules are not referenced by the DMG's guidance. The game does not appear to expect that the multiclassing rules are getting involved here - your monster should get all the class features that such a character gets, which includes whatever proficiencies a normal first level character of that class should have.
In the specific case of your advancing an Apprentice Wizard by adding wizard class levels to it, despite the fact that it looks very similar to a PC wizard already, it is not a PC wizard - it is a "monster" with wizard-like spellcasting. Adding a level of wizard to it grants it the benefits of being a first level wizard on top of what it already had - there's no affordance for it stacking up to produce a second level wizard. The DMG doesn't offer any guidance on how to combine the two Spellcasting features the monster now has, but it would seem reasonable to try and use the multiclassing rules about spellcasting to determine its new capacity in that case. If you add a fighter level, it gets all the benefits of being a 1st level fighter, including full proficiencies with armour and saving throws and whatnot.
At the point you find yourself wanting to add levels to the Apprentice Wizard, it probably makes more sense to disregard the original "monster" stat block and just rebuild the creature as if it were a PC using the normal PC rules. Stat blocks that seem almost directly representative of a PC class, like the Apprentice Wizard (Wiz 1) or the Archmage (Wiz 18), are basically there to save a DM the effort of building those characters by hand. If you start wanting to seriously modify them, rebuilding them fully using the normal PC rules will be more internally consistent and avoids any mechanical weirdness like that discussed above.