I agree with this, while certainly a novelty and a nice thing to have on a long flight it isn't exactly a great shower experience. Nothing wrong with taking the longer routing with a stop, but I wouldn't do it just for a chance at another shower. Two other things to consider - 1) They have moved EK205/206 to a 777 in the winter months (not sure when your flight is planned for) and 2) Even though the "experience" may be longer if you are sleeping or just enjoying a good movie or something, a stop in the middle of a long journey can be a real drag. While it is the same plane, you do need to disembark in MXP and end up in the lounge during the stopover.
Thank you so much for the insight! Yes, our flight would be in late January, and while EK 205 currently shows as an A380, if there is a chance for it to be down-gauged to a 777 that would be too much of a risk to take in this regard. We will stick to the non-stop then. Also, do you remember how tall the shower stall is? I am over 6 feet tall and I wonder if I will even fit. :-) Neither ChatGPT nor Google AI seems to know the answer...