I have EDS and am pretty heavily tattooed... most of them came out beautifully, gave me no problems healing, stayed bright and pretty, etc. However, one of them was done by a different artist because I was young and stupid, and the once straight lines are now pretty warped.
I asked a friend of mine about it years later who is a tattoo artist, and was told that a more experienced artist will stretch/move the skin occasionally while working, in multiple directions so that they can see where the design will go when you move/bend that body part, and they can compensate for that. An inexperienced tattoo artist will just hold the skin as still as possible and draw like they were drawing on paper.
That being said, that's going to be the case for someone who doesn't have EDS as well, because it's just proper technique, but that medical alert one has pretty much the same kind of warping as the one I have that was done by the inexperienced artist. It's just much more noticeable on someone with EDS