As of 1608 EST 27APR / 0.39.1, the random name option does work on my phone, the virtual keyboard is triggered but is still not populating the text field on my phone. So, potential downgrade of the issue from Critical to Moderate?
As of 0.39.1, still a no go for the Android keyboard filling out the name tag. 
Note the auto fill populating with data from my attempts to type something in, and the vacant field. Interestingly, hitting the carriage return/enter/downandleft arrow caused the red "Handle must contain" message to populate! So, it's seeing the submit command, at least?
As of 39.2, I have mixed news. Firefox Android browser works as expected! Keyboard pops up, field accepts data, etc etc. DuckDuckGo now locks up as soon as the name entry screen shows, no keyboard, not even accepting the random name button (like it's freezing when it tries to trigger the keyboard).
Wish I had a better report to give. Kicking myself for not backing up my export in the last few days, now.
Edit ; Opera behaves like I reported for DuckDuckGo in 39.1, before. Keyboard populates but does not fill the field.
2 days ago(+1)
Change is a good thing though! I'll see if I can sus out what the delta is between the two browsers. I suspect its a permission issue as duckduckgo tends to be more restrictive about that than most.
