I had this problem on ios a few months ago. The solution was as follows. This solution worked for me, it might not be the cause of anyone elses issues but someone might find it useful which is why I am posting.
I had signed up for Zwift with email address A but my subscription through the apple store was with email address B. I would sign into Zwift with email A but then one day it stopped working on my ipad and I would need to re-enter the password each time I logged in (interestingly, on my ipod and laptop PC it would still work OK). After rummaging around through the prefs.xml on the ipad hoping to find some clues I found my email address was set to email B. So I logged in using email B and it has worked ever since.
Like I said, this solution may not work for anyone else but if your Zwift login email address (email A) is different from your subscription payment email address (email B) then try logging in with your subscription payment email.