Is VOR identification automated in modern avionics?
Modern air transport and bizjet avionics do decode the audio ID and provide it digitally on the data output bus. This data is displayed on the Nav display or RMI (depending on Mode) along with the frequency. It's not as common for general aviation radios to have this capability.  

VORs that comply with ARINC Characteristic 711, VOR MB Receiver, output the ground station ID in two ARINC 429 Labels.  The first and second characters will be in Label 242.  The third and fourth (if there is a fourth) characters will be in Label 244.  Characters are coded in ISO 5 format.  

Since all modern radios are DSP based, detecting and decoding the audio tones is a fairly straightforward process.  The radios also provide an analog audio output, so the crew can select the VOR radio on the audio panel and listen to the Morse code if they desire. (I'd be surprised if you found a pilot that did that routinely.)  

ILS is similar. The ILS ID is contained in Labels 263 and 264.