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:
- Yes, the Drow takes the Attack action
- Yes, the Drow's attack uses a Light Melee Weapon
- 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
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.