RAW, since all NPC spellcasting features are class features, the dragon wouldn't gain the spellcasting trait of any humanoid it turns into. However, the Drow Inquisitor has innate spellcasting, magic resistance, a beefy multiattack, and a high prof bonus.

We can defer what exactly the drow racial benefits are, so those would stay, but what about everything other than the spellcasting trait?

EDIT: Can anyone tell me if there is any RAW explaining the concept of "NPC classes"? I see a lot of people saying that this is the reason dragons taking humanoid shapes effectively only nets them the racial traits and the CR of the form, basically nothing else, but I don't know where it comes from

Might be a question better suited to the Rules and Game Mechanics section. Also, this is going to be messy, as the actual text regarding this topic is, IMO, intentionally vague to allow for a DM to customize creatures as they see fit.

Regardless, the only restriction on what a dragon can turn into is based on the dragon's CR. Looking at Adult Silver Dragon, we have a CR 16 creature that can also be an innate spell caster in it's own right.

Short simple answer - yes, a dragon can become a Drow Inquisitor.

RAW concering NPC Classes? Maybe not limited to NPC classes and maybe not the RAW you're looking for, but Monster classes are mentioned in DMG, Ch. 9. 

Monsters with Classes

 You can use the rules in chapter 3 of the Player’s Handbook to give class levels to a monster. <snip>

I would also note that the MM intro lists spellcasting as a class feature, which may infer that the class features listed in any class build would potentially be verboten for our polymorphed dragon. Not super certain about subclass features, I'm of the opinion that they would be right out as well.

Spellcasting

A monster with the Spellcasting class feature has a spellcaster level and spell slots, which it uses to cast its spells of 1st level and higher (as explained in the Player’s Handbook). <snip>

And thus far, we have had two references back to the PHB for information concerning classes and class features so the information that you are looking for seems be there. I will state for the record that none of what I have proposed is strictly written in full anywhere, but inferred by looking in similar places throughout. RAW, not likely. Useable for a DMs purposes, absolutely.

I'm not entirely sure that your citation that "RAW, since all NPC spellcasting features are class features..." is entirely accurate. Innate spellcasting is specifically a racial feature that is granted regardless of class. Regardless, the dragon's innate spellcasting variant rule might become a hinderance to the assimilation of the Drow innate spellcasting ability.

Innate Spellcasting

<snip> A monster’s innate spells can’t be swapped out with other spells. If a monster’s innate spells don’t require attack rolls, no attack bonus is given for them.

General Spellcasting and Innate Spellcasting are different things.

General Spellcasting is a class thing. Innate Spellcasting is not. Innate spellcasting is part of the monster/NPC and nothing to do with class features. 

So yes, if the dragon used Change Shape into a humanoid with Innate Spellcasting - it can cast them. 

Innate Spellcasting

A monster with the innate ability to cast spells has the Innate Spellcasting special trait. Unless noted otherwise, an innate spell of 1st level or higher is always cast at its lowest possible level and can’t be cast at a higher level. If a monster has a cantrip where its level matters and no level is given, use the monster’s challenge rating.

An innate spell can have special rules or restrictions. For example, a drow mage can innately cast the levitate spell, but the spell has a “self only” restriction, which means that the spell affects only the drow mage.

A monster’s innate spells can’t be swapped out with other spells. If a monster’s innate spells don’t require attack rolls, no attack bonus is given for them.

Whereas Spellcasting is specifically detailed as a Class Feature (I've put in bold):

Spellcasting

A monster with the Spellcasting class feature has a spellcaster level and spell slots, which it uses to cast its spells of 1st level and higher (as explained in the Player’s Handbook). The spellcaster level is also used for any cantrips included in the feature.

The monster has a list of spells known or prepared from a particular class. The list might also include spells from a feature in that class, such as the Divine Domain feature of the cleric or the Druid Circle feature of the druid. The monster is considered a member of that class when attuning to or using a magic item that requires membership in the class or access to its spell list.

A monster can cast a spell from its list at a higher level if it has the spell slot to do so. For example, a drow mage with the 3rd-level lightning bolt spell can cast it as a 5th-level spell by using one of its 5th-level spell slots.

You can change the spells that a monster knows or has prepared, replacing any spell on a monster’s spell list with a different spell of the same level and from the same class list. If you do so, you might cause the monster to be a greater or lesser threat than suggested by its challenge rating.

So if the dragon became a Drow Inquisitor which has both types: the dragon could use the Innate Spellcasting, but be unable to use the class Spellcasting.

Change Shape. The dragon magically poly-morphs into a humanoid or beast that has a challenge rating no higher than its own, or back into its true form. It reverts to its true form if it dies. Any equipment it is wearing or carrying is absorbed or borne by the new form (the dragon's choice).

In a new form, the dragon retains its alignment, hit points, Hit Dice, ability to speak, proficiencies, Legendary Resistance, lair actions, and Intelligence, Wisdom, and Charisma scores, as well as this action. Its statistics and capabilities are otherwise replaced by those of the new form, except any class features or legendary actions of that form.

When a Dragon uses it's Change Shape ability, they can look like a Drow Inquisitor, but they get none of their abilities.

If you would like to modify that, may I suggest this, Druids have Wild Shape, and the rules are as follows:

Wild Shape Improvement

At 8th level, you can use your action to magically assume the shape of a beast that you have seen before of challenge rating 1 or lower. You can use this feature twice. You regain expended uses when you finish a short or long rest.

So then

Draconic Shape Improvement

As an Adult or older Dragon, you can use your action to magically assume the shape of any Beast or Humanoid of a challenge rating no higher than your own that you have seen before and freely use any of the abilities you have seen them use except for Legendary actions.

In my game I have a dragon who regularity takes human form, it is an ancient silver dragon who takes the form of various NPCs through the world the players are in. The only distinction are that these human forms are covered in tattoos that seem to move and writhe. These tattoos are its spell book, in my campaign this dragon, on top of its natural spell casting ability, is effectively a wizard and so can learn spells which then form symbols in its scales. This was taken from an older, I think 3rd edition version of dragons. 

In each of its forms the Dragon has the skills and abilities of that humanoid creature. It’s true sight allows it to represent as if it has the correct type of dark vision for the race it is pretending to be, (in one of its forms it once saw something long before the party did but allowed them to be caught out by it so as not to give itself away). It represents spell racial traits by having access to those spells itself and will then down cast them, effectively casting them as if it was a far weaker spell caster then it is. 

 Is this may not be RAW but this is an ancient 2000+ year old dragon who has been pulling the strings across the entire continent for centuries. It has created an academy for spell casters from across the world, it has collected within this academy the greatest collection of spell books and artifacts known. It has studied and read pretty much every one several times over and continues a drive to seek out more knowledge. The party believe each individual is an npc member of a secret organization and have watched 2 of these individuals die, each time the dragon made the death look real. 

My point is that while you might be looking for RAW answers as to how to do a thing, as the DM you can just do it, the rules are not the definitive list of all the magic in the world, the fact that more spells get added almost every release shows that. I have often given my NPCs spells and abilities outside of the rules because it makes narriative sense and my players have never complained I am playing out of the rules. If it makes the story cool don’t over think it, it’s a powerful dragon, just let it do its thing :). 

Quote from Geann >>

Change Shape. The dragon magically poly-morphs into a humanoid or beast that has a challenge rating no higher than its own, or back into its true form. It reverts to its true form if it dies. Any equipment it is wearing or carrying is absorbed or borne by the new form (the dragon's choice).

In a new form, the dragon retains its alignment, hit points, Hit Dice, ability to speak, proficiencies, Legendary Resistance, lair actions, and Intelligence, Wisdom, and Charisma scores, as well as this action. Its statistics and capabilities are otherwise replaced by those of the new form, except any class features or legendary actions of that form.

When a Dragon uses it's Change Shape ability, they can look like a Drow Inquisitor, but they get none of their abilities.

 

 
Incorrect. It literally states "Its statistics and capabilities are otherwise replaced by those of the new form" -- this means it can do what the new form can do, except for legendary action and class features. Innate Spellcasting is not a class feature.