That is an awkward, albeit grammatically correct way of phrasing it. In the scenario you provide the idiomatic phrasing would rather be "What are they giving/putting in me?" or "What are they doing to me?". It is when you use "going" that the meaning becomes a little more dirty!
In English we do sometimes say something "I don't know what has gotten into me lately" if we feel our behavior is a little atypical or unusual. Or we might say it like you did "What has gotten into me?" I hadn't thought of that interpretation of the sentence since it seemed more literal to me. I would also love to hear from a native speaker if that sentence could be used this way. Good thoughts.
I am not sure what that means, but I can't imagine any idiomatic meaning of this Hungarian sentence. So, I have to take it literally. Something is going into this person. My best guess is that it is from the personal experience of one of our contributors, who got hooked up to the wrong IV (intravenous) fluid just moments before he or she started making up sentences for the new Hungarian Duolingo course.