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The rule on Two-Weapon Fighting (PHB, page 195) states:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

Now, let's take for example a drow (MM, page 128).

Under its Actions, it can use a shortsword, which is a light melee weapon. Should said drow find itself wielding a second shortsword, would it be able to use Two-Weapon Fighting within the normal limitations of the feature?

Yes, if you want it to be able to

NPCs in D&D 5e are designed differently from, and often do not follow the same rules as, the player characters - but in this case the same rules should apply. The guidance for reading monster stat blocks states:

When a monster takes its action, it can choose from the options in the Actions section of its stat block or use one of the actions available to all creatures, such as the Dash or Hide action.

By a strict reading, the weapon attack options described in the monster's stat block are not uses of the Attack action - they're special actions that the monster can use, so if it uses one of those actions it's not taking the Attack action and so can't qualify for two-weapon fighting. (Functionally, these options are usually equivalent to taking the Attack action and making an attack with that weapon, and in most cases I'd personally rule as if they were the Attack action for situations where that's relevant, but that's not the rules as written.)

However, the monster explicitly retains the ability to use actions which are available to all creatures - including the Attack action - so it can take that action and make an attack with a weapon it's holding and then invoke two-weapon fighting just like a PC can.

Where things get a bit funky is when the Multiattack action is in play. A monster with Multiattack can make several attacks with a single action - which is much the same as a player character using the Attack action with the Extra Attack feature - but it is not the Attack action, so a monster that uses Multiattack cannot use those attacks to qualify for a two-weapon-fighting bonus action attack. A monster with Multiattack is probably better off using that than trying to use the two-weapon fighting rules though, since its Multiattack almost certainly has better stats and doesn't use up a bonus action.

Yes.

Two-weapon fighting rules apply to monsters per the MM's guidance on monsters and what actions they can take.

When a monster takes its action, it can choose from...one of the actions available to all creatures...as described in the Player's Handbook. (MM page 10.)

Further guidance on the next page of the MM specifically references the attack section of the PHB which contains the rules for Two Weapon Fighting.

Yes, in the example given

The Drow you mention can take advantage of the two-weapon fighting rule if it has two short swords.

When a monster takes its action, it can choose from the options in the
Actions section of its stat block or use one of the actions available
to all creatures, such as the Dash or Hide action, as described in the
Player's Handbook (MM p.10).

You quoted the two-weapon fighting text; let's check its requirements:

1. Yes, the Drow takes the Attack action


2. Yes, the Drow's attack uses a Light Melee Weapon


3. Yes, if you modified the Drow by giving it a second short sword, then it has an another Light Melee Weapon in its other hand

There you have it; the Drow can get a second attack if he has a second shortsword.

NOTE: The attack action lets you draw one weapon as part of the action. If your Drow starts his turn empty-handed, he'll still get only one attack on that turn. He can draw (and use) the second short sword as part of his attack on his next turn.

ALSO NOTE: If you want to be cheesy, a Drow would be complying with the rules if he grabbed a short sword off the ground (as an "Interact with an object"), then drew a second short sword as part of an attack action. For a CR 1/4 mook, this Drow is getting really industrious.

This might affect the Drow's CR

The CR 1/4 Drow stat block (Monster Manual p.128) lists actions for a typical Drow-on-the-street. This stat block implies that, given the nature of Drow society, an average non-soldier Drow will have chain mail, one shortsword, and one hand crossbow.

That being said, as GM, you can modify a Drow by giving it an extra short sword. For that matter, you can also change its armor or change any of its equipment. You can even declare that an enclave of Drow are all infused with demon blood that lets them shoot lasers from their eyes. The stat block is just a time-saving summary; it shows a typical specimen so you don't have to calculate combat stats for every monster the party encounters.

Feel free to tweak an existing creature to make it into something more
useful for you... Keep in mind that modifying a monster, including
when you apply a template to it, might change its challenge rating (MM
p.6).

Consult the Dungeon Master's Guide to figure out whether a change affects a monster's CR.

This is mostly limited to Low-CR Humanoids

Non-humanoids tend to use natural attacks, while high-CR creatures tend to use Multiattack, neither of which works with Two-Weapon Fighting.

Two weapon fighting does NOT work with natural attacks (claws, fangs, etc.)

The two-weapon fighting rule requires the creature to attack with a light weapon and to be holding a light weapon in its other hand. Most natural attacks are listed as a "melee weapon attack", but that's just to distinguish them from "melee spell attacks". Claws, fangs, tentacles, etc. are NOT an attack with a weapon, let alone a light weapon.

Two weapon fighting does NOT work with Multiattack

The two-weapon fighting rule lets you make an extra attack as a bonus action when you take the attack action.

Multiattack is its own action. A creature using Multiattack is NOT taking the attack action; rather, it's choosing the "Multiattack" action listed in its stat block.

Monsters that swing multiple weapons or that swing a weapon multiple times do so by listing "Multiattack" in their stat block. Again, a stat block is a summary of how a monster operates in combat.

For example, a Marilith has six arms (and a tail). On her turn, she can draw and attack with six swords (and her tail). Why? Because she has a Multiattack action that says so. She isn't using the fighter's Extra Attack ability or Two-Weapon Fighting or the Dual Wielding feet. The stat block summarizes her training and her demonic nature into the fact we actually care about: She attacks with all six swords (and her tail).

The Bottom Line

Stat blocks are a summary of what a typical specimen of a monster can do. They're a convenience; planning an adventure is hard enough without the GM having to figure out every equipment permutation on every villainous minion a party might encounter.

Any creature capable of holding two light weapons is capable of swinging both light weapons. This fact is accounted for on stat blocks by two things:

Low-CR mooks are limited to possessing the weapons on their stat block because weapons cost money. Joe McPeasant doesn't get two short swords for the same reason that he doesn't get a suit of plate mail.

Creatures that are actually GOOD at swinging multiple weapons and that tend to POSSESS multiple weapons generally say "Multiattack: [attacks with both of its weapons]" in their stat block.