If I understand what the article is saying, using DTMF tones like this does not offer anonymity for the patients on the other end of the phone call. Simply recording the DTMF tones would be enough to reveal the patient's true phone number. Perhaps there was some sort of server-side mapping scheme in use to mask the numbers and the DTMF tones were truly just for data transmission, but no such mention was made in the article.
The patient's phone number is visible to the nurse and you can see that in the app screenshot. Their goal was to mask the nurse's mobile number while having the app work offline and this solution achieved that.