Call centre agent login with answer call by handset

Yes.

Two common approaches I see with this are

  1. To have people unregister and register the extension when they get to the desk. Similar to logging in and out.
  2. Another approach is to use FreePBX in Device and User mode. This is fringe, and lacks robust support, but will give you an experience where there is a desk extension and an agent ID you can login with.

https://wiki.freepbx.org/pages/viewpage.action?pageId=5242941