Right, CLID and ANI are definitely not the same.
I'm sure there's some variation based on the switch manufacturer or software version at the CO, but it is definitely possible to confuse E911 services by misconfiguring your PBX. IIRC, without configuration (if your PBX sends no data) the network tags the call with 000-000-0000, which might trigger some kind of subscriber table lookup.
But it's also possible to send different ANI based on your originating extension. I had different ANI configured for my and a few others' home VOIP phones (connected by T1s to the PBX in the office, itself connected to the PSTN by a PRI into the router), so that a 911 call from our houses wouldn't summon an emergency response to the office in the Financial District.