I'll lay out the scenario just to give a bit of context:

I wanted to create a PC that dual wields shields as his weapons. He has sworn off bladed weapons for TRAGIC BACKSTORY reasons. 

Obviously dual wielding shields is suboptimal, but that's ok for me. I know that I don't get the additional AC from the second shield (can only benefit from one shield) and they are treated as Improvised Weapons, doing 1d4 bludgeoning damage.

If I take the Tavern Brawler Feat, Dual Wilder Feat, and play a Variant Human Fighter with the Two-Weapon Fighting Style, I believe that:

• I can add my proficiency bonus to attacks with the shields (Tavern Brawler)
• I can use my bonus action for the Two Weapon Fighting attack (Dual Wielder feat taking away the "light" requirement for weapons for TWF)
• I can add my ability modifier (STR) to the damage of my bonus action attack (Two-Weapon Fighting Style)
• Get an additional +1 to AC due to the Dual Wielder Feat.

Therefore at Level 7 I could attack 3 times per round (Attack, Extra Attack, Bonus Action TWF attack), and add proficiency to all attacks and STR bonus to all damage. 

However, this DOESN'T WORK if Improvised Weapons don't count as Melee Weapons. 

Dual Wielder says:

• You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.
• You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren't light.

Two Weapon Fighting (the rule) 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. 

The Two-Weapon Fighting Style from Fighter merely removes the "light" requirement.

So is the character build I have laid out RAW? Or do Improvised Weapons not count as "melee weapons" for interacting with Feats, Class Features, etc.

https://www.dndbeyond.com/sources/dnd/phb-2014/equipment#ImprovisedWeapons

Some GMs might stop at the two shields thing.

I would stop you on the part where a shield is NOT a Light weapon even if you have tavern brawler.

However there are some cool images on the internet for them.

All this seems legit. I believe the official ruling is a "melee weapon" is any weapon you use in melee using your melee attack bonus. So, like, while you could use a bow in melee, you'd be using your ranged attack bonus, making it a ranged weapon. But if you improvised and hit some dude with your bow, then (aside from ruining your bow) you would use your melee attack bonus and it would count as a melee weapon.

Honestly, I'd talk to your DM about the damage of the shield. The rule specifically states it deals 1d4 damage if the object "bears no resemblance to a weapon." I dunno... maybe you could make a case that it bears a resemblance to a weapon? In 3e, you could put spikes on your shield and it would count as an exotic weapon. Might be a case you could make there, not sure.

Quote from Wysperra >>

Some GMs might stop at the two shields thing.

I would stop you on the part where a shield is NOT a Light weapon even if you have tavern brawler.

 You are correct they are not light, but the Two Weapon Fighting Style from my Fighter class removes the "light" requirement. 

You still have to attack with a Light weapon first.

Enhanced Dual Wielding. When you take the Attack action on your turn and attack with a weapon that has the Light property, you can make one extra attack as a Bonus Action later on the same turn with a different weapon, which must be a Melee weapon that lacks the Two-Handed property. You don’t add your ability modifier to the extra attack’s damage unless that modifier is negative.

Quote from MaximusArael020 >>

Quote from Wysperra >>

Some GMs might stop at the two shields thing.

I would stop you on the part where a shield is NOT a Light weapon even if you have tavern brawler.

 You are correct they are not light, but the Two Weapon Fighting Style from my Fighter class removes the "light" requirement. 

 No it doesn't. It only lets you add your ability modifier to damage made with your extra attack from the Light Weapon property.

Edit: Did you mean the 2014 Dual wielder feat?

In 2014 the Dual Wielder feat allowed you to make an attack with a one-handed weapon and then use your Bonus Action to attack with a different one. However, you're straining RAI to claim the standard effects of a shield and the +1 AC of 2014 Dual Wielder.

Yup, sorry that's what I meant. Got my things switched around, my bad.

Not going into all the details for the first post, but regarding the general question, "Do improvised weapons count as melee weapons for interacting with feats such as Dual Wielder, etc.?", depending on the specific object being used as a makeshift weapon, I'd say a DM could rule that they do, based on the rules:

Weapon Equivalents. If an improvised weapon resembles a Simple or Martial weapon, the DM may say it functions as that weapon and uses that weapon’s rules. For example, the DM could treat a table leg as a Club.

For the 2014 rules:

[...] 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. [...]

I also remember a few threads about using improvised weapons for different purposes:

- Using a shield as a weapon

- Does a Shield cancel out Dueling Fighting Style?

- Using Shillelagh on other wooden objects