You can create a new FullContact account if you 'Sign in with Google' (Google SSO) and this will automatically connect your Google Contacts to your FullContact account. If your Google account is already connected to an existing FullContact account, this will direct you to your existing FullContact account.
If you already have a FullContact account, you can also sign in with the account email address and password if you click on 'Sign in with email'. This will open a screen where you can type in your account details.
In FullContact for Gmail, you cannot create an account / sign up with a separate email address + password. To create an account with a separate username (email address) and password for your FullContact account, please visit FullContact Web App at: