Approve on behalf using the admin role user is possible.
The system will prompt an approver authentication window when approved records from the admin role.
The "Approvals with e-Signature" plugin is activated on the instance, so you see this behavior.
The behavior is as expected because of the activation of "Approvals with e-Signature" on the instance.
In case if you don't want the Approver Authentication to be shown when approving on help, then there is a way to De-activate e-signatures.
Set Enabled to False on any tables where e-signatures are no longer required, from the below list:-
This will disable the functionality of showing Approver Authentication popup when approving a record by a different user.