aramis erak
Legend
A shield is not a weapon with the Light property.
Only weapons with the Light property can be used to make an off-hand attack, and even then, only if the primary weapon is also Light.
Actually, the rules don't restrict that way. You can use your off-hand with ANY weapon.
One only get the bonus action to strike if
(A) you have a light weapon ready in each hand
(B) primary attack is a light weapon in one hand
(C) secondary/bonus attack is a light melee or light thrown weapon in the other hand.
Nothing in the rules prevents you from wielding a pair of broadswords. You won't get a bonus action with them, but you can wield them both.
The Dual Wielder feat changes the conditions to
(A) You have a light weapon ready in one hand, and a 1 hand weapon in the other
(B) primary attack is the 1 hand weapon (which may or may not be light)
(C) Secondary/Bonus attack is the light melee or light thrown weapon
No where is handedness stated in the rules.
So, my right handed goon carrying a shortsword in the right, and a hand crossbow in the left, can choose to make his primary attack (the one using his action) with the crossbow, which is light, and that triggers the bonus, which is his shortsword in the right hand.
The rules don't require proficiency for the primary attack.
So, a shield bash, which I agree falls under improvised weapon, is effectively part of dual wielder already.
You make your shield bash the primary, and your rapier the secondary... Bash, bash, bonus-stab at level 5...
1d4+StrB + 1d4+StrB +1d8 potential...
And that two bashes and a stab seems very-much like how being harp-sealed* by Sir Kylson, Sir Alden, Sir Donnan, or Sir Georg felt... Real World Shield proficiency isn't just holding it in place. It's an agressive wearing down of your opponent's ability to hit you. It's a left-handed defensive weapon.
*SCA reference - somewhat hyperbolic - "being clubbed repeatedly like a baby harp seal" according to Sir Kylson. After the first time or two, you work on shield bashing... because, even tho you aren't allowed to hit THEM, you are allowed to shield bash their shield and weapons in SCA Heavy. And if you don't aggressively work that shield, any of those 4 would harp seal you.