Hello Stephen,
Thank you for reaching out to the Microsoft Community and I'll be glad to assist you today.
I understand that you've accidentally removed your account from the Microsoft Authenticator app on your Chromebook. When you try to set up authentication on the device, it will prompt you to confirm your identity through the Authenticator app itself, which is no longer available. However, you do have alternative options for adding the Microsoft Authenticator to your device again.
- Do you have any other options to verify your Identity?
By default when you no longer have the Mobile device or Authenticator app to confirm, you will have other options to confirm your identity (i.e.) using your alternate email address or Phone number. ( I have tested from my device, and was able to setup authenticator using this option)
E.g. see Image 1-

(or) if you have already backed up your account credentials to the cloud, You can recover your account credentials from your cloud account.
https://support.microsoft.com/help/bb939936-7a8...
Also, please note that depending upon the type of account affected, (outlook.com account, Hotmail account, or professional/educational account) the recovery process may vary. For educational or organizational accounts, you can reach out to the IT administrator to change the MFA. If it is a personal account, check if you have any other options to sign in to your account.
If you do not have any other options to verify your Identity, go to https://account.live.com/proofs/manage/additional, to turn off two-factor authentication (or) replace or add the security information.
Related article for your reference:
https://support.microsoft.com/help/bf2505ca-cae...
I hope this information helps. If you have any questions, please let me know and I will be happy to help you further.
Best regards,
Gunasekar N