Ah, I see. Unfortunately that's an aspect of the way the PBX is setup, and nothing to do with how caller ID works specifically. Most extensions don't have a DID (direct inward dial [1]) number, and unfortunately many inbound routing setups (especially for call centers) have no way -- or at list no advertised way -- to get to a spot where outside callers can dial an extension. For most PBX systems (eg, freepbx [2]) it's an option whether to allow direct extension dialing as part of the IVR.
[1] https://en.wikipedia.org/wiki/Direct_inward_dial
[2] https://wiki.freepbx.org/display/FPG/IVR+Module+User+Guide#I...