How to push to other origins using Azure Pipelines?
While I haven't tried this, I suppose a solution would be to have a regular pipeline (this should work in either build or release pipelines) that would check out your repo (as it's default first step) then some scripted command that adds the desired remote and pushes to it.
Some pitfalls I could see you running into:

issues with the state of the agent work folder based repo prior to a push; but that should be reconcilable with the appropriate Git commands. If the target remotes (i.e. backups) are never being committed to by anything other than the pipeline, you hopefully won't run into conflicts
arbitrary branches: I'm not a git expert, but I think there's a way to push all branches to the remote
authentication: you'll obviously need to provide appropriate authentication to the target remote(s). Credentials shouldn't be stored in the scripts in plain text. Use secured pipeline (or group based) variables
repo size: a huge repo will need to be cloned into the ADO agent workspace. Not a problem as long as you have the disk to support it.

Given the needs you listed, it seems like you should consider something like AzureDevOps services in the cloud, or private GitHub repos (recently became free), etc. Why manage your own backups if the cloud can do it for you?