It is possible to login using GitHub enterprise from VS Code without using personal access token or SSH keys. User can login by SSO if session already exist in browser else will need to sign in using their credentials. One of the ways to achieve this is to install Git Credential Manager. Git Credential Manager (GCM) is a secure Git credential helper built on .NET that runs on Windows, macOS, and Linux.
Compared to Git's built-in credential helpers (Windows: wincred, macOS: osxkeychain, Linux: gnome-keyring/libsecret) which provides single-factor authentication support working on any HTTP-enabled Git repository, GCM provides multi-factor authentication support for Azure DevOps, Azure DevOps Server (formerly Team Foundation Server), GitHub, Bitbucket, and GitLab.
For installing GCM based on OS follow the given link:
https://github.com/GitCredentialManager/git-credential-manager
Once GCM is installed and you try to clone a repository you will be able to see Enterprise login page. This can be achieved by using other Git Oauth Authorized Apps such as GitHub CLI.