Because Danes are routinely cropped, this is an excellent time to tattoo, however, anesthesia is not needed for tattooing. There is an English Springer Spaniel breeder and a Pug breeder in my area for whom we routinely do Puppy Aptitude Testing at 49 days of age. Following the testing, each dog is tattooed. Further, we regularly offer tattooing at obedience classes. Dogs come out of class, are tattooed, and return to class with no loss of attitude.
Skin varies according to the individual dog.
With a dog the size a Great Dane will become, it is necessary to consider that possible stretching of the tattoo may occur with growth. Therefore, it is advisable to make the tattoo on a 7 week old puppy as small as possible. Some tattoos stretch minimally, some a little more. My policy is to guarantee the legibility of the tattoo. This means I will touch up any tattoo I do should it ever be unreadable.