Using the SSH from the browser window lets you use SSH to connect to a Compute Engine virtual machine (VM) instance from within the Google Cloud Console.
Compute Engine manages your SSH keys for you whenever you connect to a Linux instance from your browser, creating and applying SSH key pairs when needed. You cannot manage the SSH keys that are used to connect from the browser. Instead, user access to connect from the browser is controlled by Cloud Identity and Access Management roles [1].
To connect through the browser, you must be a project member who is a compute instance admin [2]. The “root” user in the VM is an internal user for the VM, the "root" user is not a project member here. Google doesn't have any access to the VM as the “root” user of the VM.
After you (project member) have been granted access, connect to a Linux instance directly from your web browser in the Cloud Console and could transfer file [3] to the VM and then you could copy your files as required path.
[1] https://cloud.google.com/iam/docs
[2] https://cloud.google.com/compute/docs/ssh-in-browser
[3] https://cloud.google.com/compute/docs/instances/transfer-files
sudo bashto get elevated privileges. Logging in as the root user is blocked by default. To get around that will require modifying the system's security. - John Hanley