“Was it really common for people to keep their modem under their phone”

Yes. This stems from how they were originally implemented – they had a speaker and a microphone, and you sat the handset in it – speaker to microphone and microphone to speaker. The modem modulated data out of the computer into sounds, and played them into the microphone of the handset, and demodulated the data out of the sounds played back by the speaker in the handset back to the computer.