Looks like an allergy to me. I'm allergic to fabric softener, and that is exactly what my skin looks like when I'm exposed to it.
That first picture after tattooing where its super red... was it red like that before the artist started their work? Tattooing on top already aggravated skin probably wasn't the best idea, but based on your followup picture it appears to be getting a little better. Try taking a benadryl or something and see if the redness and itching gets better. I've never had a tattoo look like that, and my itching never starts until the peeling phase.
Any medical advice you get from a forum could be bogus and likely won't put your mind at ease anyway. I would talk to a doctor to rule anything bad out, and to put your mind at ease.