Of course business man is a character trait. The very fact they describe how he runs his business akin to someone who runs a very successful corporation is giving you an analogy of how he runs his business. That's describing traits of the character in how he does things.
It's like when someone describes Joker as an anarchist, that's a character trait. It describes what they do and how they do it, just like Letoker being a business man akin to one that runs a very successful big corporation.
And again, the fact that he runs a business successfully describes the character's actions and not his character/personality. It doesn't describe
how he acts or thinks, his views or opinions, the other things he does, or why he does what he does in his business and outside of it. It doesn't even describe how he successfully runs his "business". The fact that an analogy was drawn between they way he runs his business and the way the head of a successful corporations runs their business doesn't describe exactly how he goes about doing so, nor does it mean he'll be involved in boring, legitimate business dealings or anything like that. It just means that he's efficient and effective at what he does, and since The Joker has always been shown to be extremely intelligent, calculating, and cunning, that is not a character departure and should not come as any surprise.
In TDK and regardless of his ultimate intention, The Joker was successfully robbing banks and stealing money from the mob. That doesn't mean one of his defining character traits was being a "bank robber" or "thief". It was just one thing he did, and one thing he did very well. Those actions did not define the character or his personality.
Him being flashy, crazy, or a killer doesn't alter the fact Joker is doing something that's totally out of character for him.
But if The Joker controls some kind of criminal organization and still does all the kind of things we'd expect The Joker to say and do, how does that make him totally out of character?
Having those character traits doesn't make any difference. The very fact Joker is running a night club (or clubs) is for a start a huge deviation. Since when has Joker ever shown any interest in running something as boring and mundane as that?
That to me is something that already makes the character seem totally off in who he is. Ignoring the awful Joker design, and speaking strictly as a character, who here can honestly say they equate Joker to be a night club owner and business man? Nobody, because it's not who the character is.
Since when has The Joker (with a lengthy criminal career behind him) ever been in a Suicide Squad movie as a seemingly peripheral character? Why must something have explicitly occurred or taken place in the comics for it to have the right to be in a film adaptation? Who's to say he would find his criminal business to be boring or mundane? It's not like the guy's going to be sitting in an office, pushing papers, and running some kind of legitimate company.
The Joker's been around for a long 75+ years, in which he's been many things and done many different things. His most famous nickname is the "Clown Prince of Crime", partially because that's what he is -- a criminal. He's not just a typical criminal of course, but rather, he is a criminal mastermind and no crime has been beneath him -- murder, theft, kidnapping, terrorism, vandalism, torture, pranks, etc. Theatricality and spectacle are key elements in everything he does, as are all of the personality traits I mentioned previously. Based on everything we've seen so far, it appears most or all of those elements are seemingly intact and that he enjoys what he does.
And so the question remains. If those elements are intact with this Joker, how is the character totally off? Because we know he's a gangster who makes money via crime, has power over part of the criminal underworld, and doesn't magically have unlimited wealth or resources to live and do as he pleases?
Penguin could still be a vain, greedy, bird loving villain, but if they started having him dabble in schemes of trying to patent fish with his face, or break people's minds to prove they're all crazy deep down, or just wreak anarchy in Gotham to prove a point, that would be a totally off characterization for him, regardless of how much he was still the pompous, elegant, bird loving, umbrella wielding Penguin in personality.
That's true, but not close to being the equivalent of what we're talking about. This Joker isn't adopting a classic Penguin scheme or plan. The idea of a criminal making money, or having a club/business as a front for illegal exploits, isn't exclusive to The Penguin. And once again, the simple fact that this Joker seems to benefit financially from some of his exploits doesn't automatically make him ANYTHING like The Penguin. They are still intrinsically different at the core-level, especially in terms of personality and characterization.