As I see it the companies are well within their rights to choose employees who reflect their corporate image. In aviation, the image - as previously stated - is usually quite conservative.
Personally, I have however no problems with tattoos at all, as long as they're not visible under work wear. Piercings and visible body modifications on the other hand kinda feel unprofessional (and even sorry to say a bit childish).
I've been in one company medical and I had to strip down to my undies, so had I had a tattoo it would probably been noticed there.