I assume AsusWRT behaves the same for the different routers, for my RT-AC68U the captcha appears after two failed logins.
After a proper login the Captcha is gone for the next login and untill the next two failed logins.
You can also assume: if you get the Captcha, someone tried and failed to login your router.
Just ran into this today with the latest firmware for AX11000 WTF... Problem is every time I try to login it says " captcha is wrong". So now I can't login to my own router? Simple fix, I just rebooted it and now its fine.. Guess its not on "initial" login just when multiple attempts fail like when a keyboard has a stuck caps lock or shift key isn't working and a password has caps..
I can't get past the captcha it keeps saying its wrong, i am getting a little annoyed i tried using chrome and FF and not getting in that sux that means security is out the window since i cant check my logs or make changes to anything good work devs when i buy a new router ill fix this one with a hammer and never never never install this crap fw again!!!