Using the New Outlook for Mac for personal email and calendar
Sorry to hear that you are having trouble with Outlook repeatedly asking you to sign in. Since you have already tried removing an Outlook-related item from Keychain, I would like to check a few details before having you make any more changes.
Could you please let me know who provides the email account? For example, Gmail, Yahoo, or an address from your Internet/email provider? You do not need to post your full email address.
When Outlook asks you to sign in, does it show any error message, or does the sign-in window simply keep coming back? If possible, please share a screenshot without including your email address or other personal information. Can you sign in to the email provider's website with the same email address and password? Did you recently change the email password or enable two-step verification?