Race is relevant because it's the point of the joke (he's racist, but his racism is foiled). Gender isn't relevant. The picture is what makes this dude a lunatic. If I saw this shit and I worked with him I'd forward it to HR with the note "hey just file this one away, you don't need to act on it right now but you're definitely gonna want this receipt sooner or later".
What in the joke makes him racist but not misogynistic?
There’s nothing in the joke saying she’s either the only black person or the only woman in the bar. In fact you could say that there’s more pointing to him being misogynistic in that he says “this woman” and not “this black person”.
Okay, that’s a fair point as to that comment not making it more sexist than racist, but not for it only being racist. I don’t see anything in the story pointing to race over sex.
How is gender and race at all relevant for this joke?
The story itself isn't a joke. The linkedin salesman guy wrote a stupid story about an unrealistically obnoxious rich guy who trips over his own racism.
How is that picture appropriate for a professional networking site?
Maybe he's targeting people who look like that. The [guy who is rude to my target demographic] is quietly pwned by [my target demographic] (Now they will buy my product).
Also, what did the dude google to find that picture?