To my knowledge, simply granting a user ownership rights over a call handler isn't sufficient for them to login to the admin UI; they must also be assigned an admin role. Granted, the admin guide seems to claim otherwise:
Call handler owners can select a different call handler greeting or record the call handler greetings from the System Call Handlers > Greetings page in Cisco Unity Connection Administration, or they can use the Cisco Unity Greetings Administrator to do so by phone. (For more information on recording greetings and using the Cisco Unity Greetings Administrator, see the “Managing Recorded Greetings and Recorded Names in Cisco Unity Connection 9.x” chapter.)
Perhaps someone else can close this descrepency; however, the way I have always dealt with this is to use Greetings Administrator. It's an IVR conversation that allows a user to login, enter the extension of the call handler, and then manage the greetings. This is probably a better fit anyway since you can attach a DID to it and the user can call in from wherever they happen to be.
http://www.cisco.com/c/en/us/td/docs/voice_ip_comm/connection/9x/administration/guide/9xcucsagx/9xcucsag150.html#59457
