From Basic:
Improvised Weapons
- Sometimes characters don’t have their weapons and have to attack with whatever is close at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a
frying pan, a wagon wheel, or a dead goblin.
- In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club.
- At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
- An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type
appropriate to the object).
- If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a
normal range of 20 feet and a long range of 60 feet.
Two-Weapon Fighting
- 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.
Shields.
- A shield is made from wood or metal and is carried in one hand. Wielding a shield increases your Armor Class by 2. You can benefit from only one shield at a time.
Light.
- A light weapon is small and easy to handle, making it ideal for use when fighting with two weapons.
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A shield is not a weapon, but can clearly be used as an improvised weapon, since anything can.
"LIGHT" is a weapon property defined by how the weapon works in 2WF. It is not a general property defined by the weight or an object. It is narratively and generically linked to a weapon's size and shape, but the only thing that really determines whether a weapon is light is if it has the LIGHT tag on it in the weapon table. Objects that aren't weapons don't even contemplate the possibility of having this tag, even if used as improvised weapons. Only a generous DM can decide than an object used as an improvised weapon works as "LIGHT" for 2WF, but even an object that is "light in terms of weight" isn't necessarily "LIGHT" for 2WF.
The off-hand attack as bonus action of 2WF is possible exclusively with a "LIGHT" weapon.
Conclusions:
> Without special abilities (e.g. feats) you cannot attack with a shield as part of 2WF.
> You can attack with a shield using your normal attack action, and doing so doesn't make you to lose the AC bonus.
You take pains to detail the requirements of TWF. It requires an attack with a (light) (melee) (weapon) (in one hand), which then opens up a bonus attack with a (light) (melee) (weapon) (in the other hand).
You pointed out that the (light) requirement would fall away if you have the Dual Wielder feat, and the rules for TWF already let you replace the (melee) with (thrown), but there is nothing that takes away the (weapon) requirement.
The improvised weapon rule allows you to attack with objects do not have the (weapon) tag. Those rules do not give a (not weapon) the (weapon) tag any more than they give the (light) tag or the (finesse) tag.
Since shields (and any other improvised weapons) are not (weapons), they do not qualify to be used to execute a TWF attack, by your own logic.
But what about Jeremy Craword's logic?
Q: A shield as an improvised weapon with the Dual Wielding feat: Does the feat's +1 AC stack with the shield +2 AC that round?
A (from Crawford): Dual Wielder is meant to work (RAI) with a melee
weapon or an equivalent,
not something like a shield.
Q: Um, you're the one who previously Tweeted that a shield can be used as an
Improvised Weapon! How's that not a melee weapon?
A (from Crawford): Yes, a shield, like many things, can be used as an improvised weapon. This has no bearing on the design intent of Dual Wielder.
The improvised weapon rule exists to allow you to attack with objects that are
not weapons; it doesn't mean that they actually
are weapons. The word 'weapon' has a meaning in the game, and interacts with other rules. Some spells require them to be cast on a 'weapon', fighting styles benefit 'weapons', the TWF rules require 'weapons'. Just because the improvised weapon rule allows you to pick up any old object and whack someone with it does not mean that the object itself has changed into an object designed to be used in combat, which is what the word 'weapon' means in the game.