Hello, I would say that the agents should use a greeting that matches the line / service. However, callers may not actively listen - i know this because I have had agents move from different accounts and accidentally use an old greeting for a different company - and quite often the caller doesn't even notice (but the agent and those around them do).
This will add pressure to the agents - but presumably their telephone or PC indicates which line they are dealing with? - that being the case they need to establish the habit of physically checking that and amending their greeting to the correct department / service.
I'd suggest a standard opening such as Good Morning / afternoon / Evening. Thank-you for calling XXXXXX~department Name XXXX - you are speaking to (Agents name), may i take your name please.....
Customers (including me!) do get anoyed with IVR's and as such I would never assume they have selected the correct line, if they have just randomly chosen an option in the hope of actually speaking to a person then by using the service name in the greeting some will then quickly say - actually I'm trying to get through to...... and the agent can ideally transfer them - or advise number and IVR options...
