I accidentally replaced my authenticator backup with new one.

Sorry but if the backup is overwritten there's no other recourse. Some general info; You can restore from backup (assuming there was one) but make sure no accounts have been added to the newly install app. Then sign on with recovery account to do the restore.

You can recover your account credentials from your cloud account, but you must first make sure that the account you're recovering doesn't exist in the Microsoft Authenticator app. For example, if you're recovering your personal Microsoft account, you must make sure you don't have a personal Microsoft account already set up in the authenticator app. This check is important so we can be sure we're not overwriting or erasing an existing account by mistake.
https://support.microsoft.com/en-us/account-billing/back-up-and-recover-account-credentials-in-the-authenticator-app-bb939936-7a8d-4e88-bc43-49bc1a700a40

--please don't forget to upvote and Accept as answer if the reply is helpful--