What a weird analogy. Putting a logo on any product is called branding. So a logo on a prosthesis is just having a branding on oneself. Which is a much more powerful analogy, even at the cost of being a tasteless pun.
When you then have your prosthetic then being an advertisement, it can have an impact on your relationship with the device.
I don't want someone branding my arm and I can see why others would feel the same.
That's exactly the point of my analogy?
I think tattoo does imply more of a permanence and familiarity in regards to the relationship someone would have with a tattoo on a part of their body they're born with, but I do see your point as well.