I think a cool idea would be a fighter that only uses shields. Just two shields that they wack people with so yeah. I know that your AC can only benefit from one shield at a time but yeah.

I tossed up a similar idea, though they were a Goliath barb, based of Baxia from Mobile legends. I would say that shields should do a d6 damage, as that isn't game breaking, though it is up to your DM (but who is gonna say a giant metal shield does a d4). But yes, the idea is cool.

Well in Volo's Guide to Monsters there is this Fire Giant Dreadnought that attacks with shields and is does 3d6 damage if it fails a save. So you could have it were you have to make a save. I would also say 1d6 dmg.

RAW, a shield is an improvised weapon (except one or two magic shields, I think) so unless you take tavern brawler feat, you wouldn’t add proficiency bonus to attacks. It’s certainly not a light weapon, so you’d also need two weapon fighting to use two of them. So after two feats, you could do a d4+str twice a round. 
As it seems like you realize, a DM might allow you to do it some other way, because rule of cool. But RAW, it would be ineffective. 

But if you're proficient in shields, then the DM would probably be more inclined to let you. And do you think that a big shield would only do a d6? I'm not trying to be rude, or that you necessarily agree with what raw may say, but making a fighter take the tavern brawler to be proficient in shields again is a bit annoying. But I guess from a rules perspective, then you are right. But as fighter, getting two feats is easier than every other class. 

Yeah I don't know how much dmg it should do. Maybe you could put spikes on the front to increase dmg.

I think RAW, it would only be a d4, if you want to houserule otherwise, I don’t think it would be game breaking. 
While I don’t usually like to analyze things too much, I will her, because why not. 
It weighs 6 pounds, which is a comparable weight to a heavy/two handed weapon. But you can only use it with one hand, so it’s going to be awkward to control. It’s attached to you forearm in such a way that you can basically only use your shoulder muscles to swing it. Enemies will know it’s coming, since you really don’t have a lot of options in what you target (you can’t really swing high or low very easily, for example) You won’t get the benefit of extending your arm fully, or really much leverage behind the blow. And since it’s attached to your arm, your target has to be close, basically the length from shoulder to elbow at max. So you have to get in very close, with something heavy enough that it should take two hands and ram it into someone who will have a very good idea of where the blow will land. If I remember my high school physics correctly, the speed you swing a thing is more important than the mass, and there’s no way, with one arm, you’re swinging it very fast. And then you want a second one on the other arm, which, again weighs more than any one-handed weapon. 
To me, a d4 for something that awkward seems about right. 

All that said, this is a fantasy game about heroes who do all kinds of crazy crap, and I don’t think it would really be game breaking to houserule that you can do it and give you that d6. Or even houserule a feat for it specifically. Probably it would be pretty fun and a memorable character, which is what really matters. 

Yeah. Maybe if  you just got one shield and used it to ram people that would work better. But I don't know.

This made me think of something like a ring of the ram, but with a shield. 

Yeah. That would be cool.

I don't know that you'd even need to houserule something; if it were me DM'ing, and you said, "Hey, I'd like to have a fighter who dual-wields shields, bashin' people with them and whatnot," I'd say, "Cool. Which one-handed bludgeoning weapon would you like to reskin as a shield?" Since we're not talking about the character gaining a double-shield benefit, there's really no mechanical difference between fighter-with-two-shields and fighter-with-mace-and-shield. Heck, say there's spikes on your shield, and I'd say, "Which one-handed piercing weapon you wanna reskin?" Because they're unwieldy, I'd say you couldn't dual strike with them as a general rule, since to gain the AC benefit you'd have to, you know, use one as an actual shield. But heck, if you wanted to dual strike with them, I'd just say you have to equip them in a different way (straps in different places?), and wielding them like that means you're overextending on each attack sequence, so you can't gain the AC benefit. Just tell me prior to each combat how you're wielding them - no different from a fighter going into combat with a shield and flail, then dropping the shield to equip a short sword for the second attack. I'd keep the action mechanics the same for all of the unequipping and re-equipping necessary to switch from dual attacking back to weapon-and-shield.

And I mean, I suppose there could be a situation when the dual-shields might make a mechanical difference...trying to gain benefits from two magical shields at a time, perhaps, or trying to give another PC one of your shields in a desperate combat situation, but as DM, I'd have jurisdiction over that, too, and could mitigate/rule on those very specific situations if they ever arose. So my vote would be to let your fighter wield a shield and a melee weapon reskinned as a shield. Have fun, yo.

Re-skinning is houseruling. As would be equipping them in different places with different straps. 

Edit: I think it could work, and it’s a good idea. But it is a houserule. 

Okay thanks that's really helpful!