My girlfriend has Body Dysmorphic Disorder, and in her experience as well as most others with BDD (according to her as well as most other informational videos I've seen) the desire to remove limbs is a very rare instance, and sometimes completely unrelated. And unfortunately, those are the only cases you hear about.
My girlfriend's main problems are with her weight and her nose. She has an eating disorder and she's very thin, but with help she's trying to focus on being healthy and not letting the BDD win. It's the same with her nose, she desires to have one that's much smaller than it is now and she considers cosmetic surgery - though she realizes herself that the BDD likely won't go away with the surgery, so she's also working on self-acceptance, which is proving to be very difficult for her.
BDD manifests differently in each person with it, but as I stated before, the instances of people wanting to get rid of limbs are pretty rare, and the instances of them actually doing it is even rarer.
Like others have said, there are similarities in the sense of "incorrectness" between people with BDD and people who are transgender, but they are very different. Body dysmorphia is VERY different from gender dysphoria. Body dysmorphia is a situation where the appearance changes in their vision to something grotesque that they have to resort to extreme measures to rid themselves of. Rather than seeing themselves in the mirror and seeing something they're not happy with, they see something ENTIRELY different.
People with BDD often develop obsessive behaviors regarding their appearance, like avoiding or constantly looking in mirrors, applying and removing makeup over and over again, and in some instances self harm in various forms. My girlfriend's obsessive behaviors involve staring at herself in any mirror she passes and sometimes going to lengths to find one (now she carries one on her at all times) and applying and removing makeup to the point that her skin is severely irritated from it. It can take her hours to get ready for a simple trip to the supermarket if she's having a particularly bad day, and if she just can't get her makeup right, she'll stay home just in order to fix it.
There are people who desire to remove limbs, and while I'm all for bodily autonomy and freedom of expression, I do believe that people who believe they need something so extreme (ONLY IF it's due to BDD) should go through intense therapy before they go ahead with anything.
People with BDD are NOT being trendy in the slightest. It's a real and very debilitating mental illness that can severely impact one's quality of life, and even drive some to suicide.
That being said, BDD and being transgender are very different. People who compare them often have severe misunderstandings of both. Some people use it as a verbal weapon, but most are simply misunderstanding the implications of the disorder and being transgender.
BDD is a legitimate mental condition that needs intense therapy to try and "correct" (and even then the rate of cure is extremely low; it's considered a chronic mental illness) so I can see why many of you feel it's offensive to be compared.
But please be aware that people with BDD aren't doing it just for the sake of it. They're victims of a terrible mental illness that will likely stick with them for the rest of their lives and often drives them to do impulsive things they wish they didn't want to do.
