Ime it has been used by some people who struggle with their weight as a way to dismiss/ excuse my experiences.
I'm naturally thin, and eat a lot. But not snacks & only when I'm hungry 99.9% of the time. This apparently makes it acceptable to comment on my size/ appearance/ eating habits/ lifestyle etc. And I'm not supposed to find this incredibly rude, in the way it would be to make similar comments about fat people, cos you know, thin privilege.
I can't possibly have ever experienced anything bad in my life, cos I'm thin. An abusive childhood can be dismissed as a cause of low self esteem when I was younger, because it can't possibly compare to low self esteem in a fat person.
If I find it incredibly difficult to find clothes on the high street I am either lying or it is my own fault for starving myself. God forbid that I point out 20yrs ago lots of shops etc had reasonable tall ranges in store, and a size 6/8 actually was small, now the tall range is a rail or 2, replaced by a plus size range, and even in stores that go down to a 6 vanity sizing renders them too big. Instead I have to listen to complaints that h&m is awful because it is one of the few larger people can't shop in, or can't get in the vanity size they usually wear.
I don't know about bias, but I doubt it personally has been a net gain for me. I've been told by quite a few people that when they first met me they assumed I'd be a bitch/ love myself/ be obsessed with diet/ have a perfect, smug life etc. And how they were pleasantly surprised that I'm not. I can't imagine that false first impression would act in my favour.
It's also very offensive to compare it to white privilege, even for those who have been treated badly through being overweight. It is not remotely on the same scale as the long history and present consequences of white privilege.