How do I understand the command airdump-ng is looking only for access points?
airodump-ng wlan0 will listen for any 802.11 devices beacons (master and managed mode) at 2.4 GHz. So, you know it is not only capturing AP beacons, your only alternative is to filter clients later using something like Wireshark if this is an issue. It is important to realise you are entering your Wi-Fi card into monitor mode (not managed mode) for this type of surveillance.
I do understand this can be an issue, so I would suggest specifying the BSSID of the AP before you start capturing and writing to a file in Airodump-ng, as the legality of intercepting traffic outside of your network is in question.
It is necessary to determine student attendance.
I do not understand exactly what you are asking here. But, you can look through all BSSIDs associated with your station (AP) MAC address in
airodump-ng or Wireshark.
Is it possible to get the MAC address of these devices, given that they are not access points?
Passively via monitor mode, presuming you are within the distance of capturing beacons of the clients, then yes! You will see this in the image above.
To present this data in a more readable or appropriate format look at the Aircrack-ng documentation and Wireshark. Although, for anything regarding capture dumps, I would recommend Wireshark for processing and filtering.