Striking the right teasing tone
I am writing about a behavior common to a close relative and a friend. They notice something about someone and then "give them the business" about this flaw.
You probably know this type of jovial person who will write this off as a normal part of being a close friend. The ones I know seem to be able to pull it off without seeming unkind. But I feel that if I poked fun at their flaws, I would end up sounding or being mean.
Am I being too sensitive? Or can we both be right?
To your general point, I'll say yes, you both can be right. There are definitely people who know how to get the tone (and humility) right, to tweak someone in a way that is clearly affectionate.
And there are definitely people who can't strike that tone and whose "giving the business" just sounds opportunistic or mean, whether they mean it to or not. Most people probably belong in this second group. Good for you for having the self-awareness to put yourself there.
I'd add a third group accordingly -- the people who think they're socially skilled and self-effacing enough to poke good-natured fun at others, but aren't.
But I'm wondering where being "too sensitive" comes in. If you mean that you don't like how it feels to be on the receiving end of "the business," then that supersedes any discussion of whether your close relative or friend has the personality to "pull it off." Someone's company is only as good as you deem it to be. You're entitled to decide for yourself whether you enjoy someone's character, sense of humor or dexterity with your flaws.
It's also your prerogative to say so, and to opt out of sharing precious time with them that's better reserved for someone -- or anything -- else.