Safari forget passwords (auto login) when open new tab

If You are browsing in Private Mode only - then EVEN if you close completely Safari - and open it again - EVEN if You do shut down/restart....... it will oper in Private Mode by..... DEFAULT!!! --

You have to close both tabs normal and private browsing mode and right click on safari icon and quit safari , if in safari preferences > General > if you have set / saved safari opens with new private window it will open private browsing mode only not a normal browsing mode .

And if private browsing mode is enabled , on the top menu bar is slightly grey and it is clearly written private browsing enabled , see this screen shot .

You can open as many tabs one for normal browsing and other private browsing simultaneously .

In first tab if normal browsing mode is opened it will save passwords and if you open a private browsing window by pressing command , shift and N keys on the keyboard in another tab ( to open a new tab press command and T ) the passwords will not autofill see this article Browse in private in Safari on Mac - Apple Support

See these articles Switch from two-step verification to two-factor authentication - Apple Support

Set your Mac to automatically log in during startup - Apple Support

What web site? More information would help.