No, when you login to your Anki account in the AnkiMobile app, the app remembers your login information and you don’t have to enter it again. (Unless you do something like manually log out.)
For example, I logged into my account on the AnkiMobile app on my phone about 1.5 years ago, used the app almost every day since then, fully closing the app and re-opening it at times, fully powering off my phone and powering it back on at times, performing a few iOS upgrades over that period, etc., and I’ve never had to re-enter my Anki username or password since that initial time 1.5 years ago.