pipx install auto-aws-sso

You need that to save your browser user data. Browser will be opened by auto-aws-sso.

auto-aws-sso --no-headless

Script reads aws sso login output from stdin and parses it. Then it opens chrome with user data dir and navigates to AWS SSO login page. After that it fills the form and submits it. Finally it waits for aws sso to confirm login and then it closes chrome. There is default headless mode so you won't see anything. If you want to see what's going on you can use --no-headless option. It will open chrome and you will see what's going on.

You need google-chrome installed and in your PATH.