Having to sign in in Google and accept the cookies everytime I open a new Tab in Safari

After you use new Tab in Safari and use that tab to login to Google,

..what happens? ~ Does the login fail afterward.?

There may be 'cookie setting' in Safari -- that only applies to one window(?)

Or to one Tab. (For my google accounts, I use Firefox browser login.)

Perhaps you may consider using Google support Help for ideas.

In searchable site, examples of Gmail question & answers follow:

• "Can't sign in to your Google Account" - Gmail Help

https://support.google.com/mail/troubleshooter/2402620?hl=en&ref

Cookies are turned off:

Turn on cookies, then try signing in again.

If you're still getting an error message, here are some possible solutions.

Try each one, then try signing in.

• Open a new browser window.
• Clear your browser's cache and cookies.
• Change your browser's privacy settings.

Click blue Help link, for general topic details on Google Mail Help.

Second more detailed results link offers more content than above.

https://support.google.com/mail/troubleshooter/2402620?hl=en&ref_topic=7065107#ts=2402624%2C2402700

• Sign in with a supported browser - Computer - Google Account Help

https://support.google.com/accounts/answer/7675428?visit_id=637393012957323663-2275232907&rd=1

..Since there are several Account services offered by google online

to further narrow the focus to topic question, can be most useful..

For other google applications, such as their browser, use help in there.

Sorry to not have an greater understanding of your non-Apple issue.

Good luck & happy computing!🐌🌻🐢