Here is the solution to solve you needing to always login to websites on iOS 13 despite enabling Keep Me Logged In with iPhone, and iPad.
How do we solve Keep me signed in Isssues?
One of the most annoying issues I had with my iPhone and iPad was that despite ALWAYS choosing, remember this device, or save login information, I always needed to reauthenticate myself every time I tried to log in to any website. I noticed this in particular with websites such as Google, Gmail, and Microsoft Fsmily settings that use 2-factor authentication (2FA).
ISSUE: When you enable “Keep Me Logged In” on any website using iOS Safari, you are required to log into the website as well as re-authenticate your 2-factor authentication (2FA).
Solution to iOS and Mac “Keep Me Logged In” not working
Step 1: Navigate to SETTINGS.
Step 2: Navigate to Safari (this should be located below iTunes & App Store and Wallet & Apple Pay section on your iOS device.)
Step 3: Scroll down the Safari Settings menu to the Privacy and Security Section.
Step 4: Disable (turn from green to gray) – Prevent Cross-site Tracking.
Step 5: Close Safari Settings and Log onto your favourite website.
Step 6: Be sure to enable “keep me logged in” or similar on the website.
Step 7: You should now be able to continue to log into the website without reconfirming your username and password. As a reminder, if you clear your cookies, log out of the website or surpass the websites normal login time limits you will be required to log into the website again.
What is Prevent Cross-site Tracking?
Every time you visit a website, it gathers data about your device—such as your system configuration—and uses that data to show you a webpage that works well on your device. Some companies use this data to try to uniquely identify your device—known as fingerprinting. To prevent this, whenever you visit a webpage, Safari presents a simplified version of your system configuration. Your Mac looks more like everyone else’s Mac, which dramatically reduces the ability of trackers to uniquely identify your device.
It appears that Safari looks at authentication websites for simplified logins like FaceBook Login, Google Login, or Microsoft Account as a cross-site and not saving any of the authentication / login information. This is requiring you to need to login in every time you visit the web site.