It's not actually called the Obamaphone. That is a nickname because of the large number of people who are now on food stamps, welfare and Medicaid under this administration, and can qualify for the government funded phones. The free phones given out by the government are funded by the "universal service fund" on our phone bills. You can usually find it on the second page of your bill. The first telephone tax was passed in 1898 to fund the Spanish American War. It stayed on the books in some form until 2006! The Lifeline program ( the free phones) was spearheaded by President Wilson iN 1913. It was expanded by FDR in 1934. It was modernized by President Clinton in 1996.
The cost to those paying their phone bill has increased significantly since 2008 which may be another reason for the nickname. You are right, they are not actually called obamaphones.
but...it didn't start with Reagan.