Translation:Are you having a shower?
Well, the English "Do you have a shower" could be referring to the equipment for taking a shower. Cf. "Do you have a bathtub." In that sense, a native English speaker might utter your sentence.
On the other hand, if the meaning is to take a shower, then it is very unlikely that a native speaker would utter your sentence in isolation. It would have to be part of a longer sentence such as "Do you have a shower every day."
But you could use the progressive (-ing) tense here: "Are you having a shower?" = "Are you taking a shower?" = "Are you showering?"
That suggests do you ever shower rather than in a short time frame - and definitely not whether you are doing it right now.
Duo was quite happy with my response, "Are you showering?" This is an everyday English expression in my household.
I would not report it, as people tend not to say 'have a shower' in the same way that they say 'have a snack,' or 'have a break.' You could ask, as ion1122 points out, if a person has a shower in their bathroom, but that is a different question, and in this case the context makes it clear that the question is about the verb 'to shower' = zuhany.
I guess the difference in zuhanyzás / zuhanyozás is also non existant? (see above Ryagons answer for the verb)
But I find also zuhanyzó (and zuhanyozó) and zuhany is already something?
What is what?