It's not stigma but it is a stereotype of autism, that is, a possible trait of the condition. But I'm one of the most sarcastic people you could meet, so is my cousin who also has Asperger's. Both of us our relaxed behaviour when we're around family or close friends can come across as immature, sarcastic like behaviour.
In regard to empathy, what they mean by lacking empathy is not the same type that people most commonly associate with the term. It's not about not
caring that some is troubled, it's not being able to
see someone is in trouble (Affective empathy and cognitive empathy respectively). It's cognitive empathy we can lack. But even then not everyone experiences this.
See this video if you want to know the full science behind what they mean about empathy; it's a good watch if you're interested in the autism and other mental conditions like I am.