There is no self identifying "Persian" ethnic group in Iran. The indigenous name for the Persian language is Farsi, no one in Iran identifies as "Fars" or "Farsi." Those who come from areas where Farsi has long been spoken will identify as "Irani" and/or whatever province they come from (Isfahan, Shiraz, Hamedan, Khorasan, etc. ) If you belong to a linguistic minority you may self identify with that group, but you will also identify as Irani. For example, my father's family comes from an ethnic group known as Lurs, who are closely related to Kurds. They have their own language, which is an Iranic language related to Farsi and Kurdish, and are also Shia Muslims. These people will proudly self identify as Lurs and Iranis, and there is no perceived conflict between these idenities. The same is true for many other linguistic minorities.
The only groups that probably identify as Persian are the Farsiwan and Tajiks of Afghanistan/Uzbekistan/Tajikistan or the Pars of Azerbaijan Republic. Also, many people from Iran who live in English speaking countries identify as Persian regardless of their linguistic origins. But, these exist outside of the modern borders of Iran.
The name Persian is one that is solely used in the West. People from the Iranian plateau have always self identified as Iranian as long as you spoke and Iranic language and come from the Iranic tradition(Azerbaijanis speak a Turkic language, but they primarily descend from Iranic speakers and have an Iranic tradition, plus they are Shia).