A Dual Shield fighter is possible, but is restrictive and fiddly.
First, when making high-concept stuff like this, it's critical to be aware of potential pitfalls of your character. Take a look at this guide to get a sense of the traps involved.
To realize this, my first preference would be to refluff some more "standard" version of the fighter into a double-shield fighter. Holding two shields does nothing for your defensive capabilities. But, with that said, the feats of superior proficiency "spiked shield" and midnight blade student make dual strike quite... interesting. You can "shield bash" two people a turn. Of course, the problem is in good positioning and actually doing damage. While entertaining, the dual strike fighter tends to be severely "one-trick pony"-ized.
As a first time player, I would instead look towards the Knight-variant of fighter. There's less "fiddly" stuff, and the idea of a heavy shield + spiked shield can serve you quite well. It's also very very hard to go wrong with Dwarven Weapon Training, with a Heavy shield and "gauntlet axe" (if you want to fit your personal style) and just say the "gauntlet axe" is a razor-sharp shield.
Two shields as "important" only really comes into play with the Snapping Tetsudo Paragon Path, a relatively uninspiring choice, unfortunately. To be fair, Dragon Magazine 385 has a significant number of "shield-as-weapon" choices, but few of them require two shields. All that being said, I would still go with knight + dwarven weapon training, and either pretend a hammer or an axe was a shield, depending if you want to be bashy or slicy. The mechanics fit better, and the thematic imagery can be just as vivid.
The bonus proficiency is a domain feature. Domains are a class feature. The multiclassing rules state that you gain the class features when you get a new level in the new class. It then lists exceptions to this with channel divinity, extra attack, spell casting and unarmored defense. Domains are not listed, so it seems that the bonus proficiency is the more specific rule and you gain the domain proficienices.
Best Answer
The ability is monster-only
The Fire Giant Dreadnought's ability is an instance of a "specific beats general" rule. Generally, a character can only gain the benefits of a single shield. The Fire Giant Dreadnought has a specific rule stating that it can benefit from dual shields, so it can do so but this doesn't apply to any monster or character without such specific rule.
No change to the general shield rules are in the Player's Handbook Errata.
House ruling the shields is always an option, but may cause poor balance especially on lower levels when sources of AC are very limited. DnD 5e is designed with the principle of bounded accuracy (see for example this answer, meaning that AC and attack bonuses scale slowly with growing level. An extra shield will put one's AC several levels ahead in relative defense, and significantly reduce the threat level of enemies who now need to roll very high (possibly even natural 20) to score a hit in the first place.