The Legend of Korra deepened the world of the avatar, fast-forwarding society 70 years into a steampunk (and borderline dieselpunk) world where technology and progress defined the age. Avatar Korra did her best to keep up with this (she was born in the boondocks of the Southern Water Tribe), while Varrick and Zhu Li were at the forefront of progress. And that even reflected in their fashion choices.
In Seasons 3 and 4 of this show, the entire world was shaken up in many ways, and nothing was the same ever again. Between Korra opening the spirit portals, Zaheer assassinating the Earth Queen and Kuvira's rise to power, change was the order of the day, and when the dust settled, Varrick and Zhu Li embodied that change and radical thinking during their nuptials, of all things.
Varrick & Zhu Li Turn Over A New Leaf
Varrick's personal arc took more than a few twists and turns, and by the end of it, Varrick had nearly redefined himself as a person, and his relationship to Zhu Li as well. This is what their wedding attire represented. Going back, when Varrick was introduced during the Water Tribe civil war, he was a shrewd, greedy businessman with a few schemes up his sleeve, and he profited greatly from the civil war unfolding around him. He even went all the way to Republic City to shoot propaganda films to fan the flames of war.
Later, after that scheme fell apart, Varrick visited the great metal city of Zaofu to slowly build himself back up, and three years later, he had joined Kuvira's Earth Empire so he could continue to build and innovate. He was asked to make a great spirit cannon and refused, so he and Bolin cut ties with Kuvira and fled.
Like Tony Stark, Varrick had grown weary of war and schemes, and he wanted to redeem himself as a hero and help fix the world. Zhu Li felt the same, and she independently turned on Kuvira and ended up on Korra's team, along with her former boss. Together, Varrick and Zhu Li helped take down Kuvira's giant robot for good. Their changes were personal, too.
At first, Varrick thought only of himself, having an inflated sense of ego, and Zhu Li put up with it, since she wanted to help Varrick invent the future (and indeed he did). But by Book Four, Zhu Li's feelings toward Varrick were more intimate than ever, and this (among other things) drove her to confront Varrick and call him out on how he took people for granted. Zhu Li and Varrick were peers and equals, not boss and minion, and Varrick was stunned at first but then began to see things in a lew light.
All his guilt came crashing down on him in an off-screen "what have I done?" moment, and he emerged as a kinder, humbler man. In fact, he returned Zhu Li's feelings, and he was no doubt greatly impressed with Zhu Li's courage to stand up to him.
They got engaged soon after that, and thus began a new chapter in their lives together. To visually represent this (aside from the engagement ring), Varrick and Zhu Li wore distinctly western-style clothes for their wedding, contrasting with the Victorian-Asian fusion fashion that surrounded them.
These two were stepping forward together into something new, and their standout suit and gown represented how they had something unique between them, and that (in a good way) they were lost in their own little world together. They've always been eccentric and off-beat, and now, they're inventing the future as husband and wife. How could they not dress for the occasion?
Varrick, Zhu Li and The World's Progress
At the end of The Legend of Korra's animated series, the entire world is undergoing serious change, with Varrick and Zhu Li being just the start. The Earth Kingdom was replaced with the Earth Empire, and now the Empire must figure out its future without Kuvira, the Great Uniter, behind the wheel. The spirits and humanity share the world, and Korra and Asami realize that they have something special with each other. Both as a pen pal and later in person, Asami was Korra's best friend and most tireless supporter, and by the end of the series, Korra and Asami formed the series' first (in release order) LGBTQ relationship.
This is breaking new ground for the franchise, and it's possible that Varrick and Zhu Li's unusual wedding attire not only represents their own relationship's transformation but also the entire world's progress into the future. Anything and everything is possible right now, and there's an air of creativity, optimism and opportunity everywhere. Why not get a little creative and stand out with brand-new wedding attire, while Korra and Asami get together at last?
A wedding is one big party, after all, and a time for people to express themselves the best way they know how. Even Bolin expresses himself by serving as the officiant, something he's never done before. Bolin is a people person, and he shows that in new ways, here. It's a time for everyone to creatively express their best selves.