Sources for this period are few and not entirely reliable. What we do know is that the Rum-Seljuk capital Konya was populated by people of diverse ethnic backgrounds, with a lot of interaction between them. The political classes under the Seljuks and Muslim principalities (beyliks) included individuals of various ethnicities. The patronage of these individuals, as well as conversions, marriages and trade, contributed to making the boundaries between ethnic groups very porous especially in the larger cities. However, the situation may have been very different in rural areas. There are references in twelfth-century Byzantine sources to nomadic Turks who were moving across the Anatolian plateau according to seasons, and they seem to have had a separate life from the settled populations.
Much later records from the Ottoman period show that there was a certain concentration of non-Muslim ethnic groups in particular neighbourhoods in urban settings, but it is difficult to tell whether this had always been the case. Some of this was no doubt the result of Ottoman sürgüns resulting in emigres from particular rural areas being settled in the same neighbourhood. Recent research suggests that the boundaries between the social realms of Muslims and Christians in medieval Anatolian urban society may have been, though not entirely non-existant, more fluid than they would be in the later Ottoman empire.
The identification as Roman (or Rumi) could be adopted by individuals of any ethnicity, and could mean different things according to context. Ways of urban life probably played a greater role than ethnicity in an individual’s identification with a particular community. A Tukish-speaking city dweller may have felt a closer affinity with Greek denizens than with the nomads.
Some sources where you can might answers to this question:
Cemal Kafadar, “A Rome of One’s Own: Reflections on Cultural Geography and Identity in the Lands of Rum”. Sibel Bozdoğan & Gülru Necipoğlu, History and Ideology: Architectural Heritage in the “Lands of Rum”, pp. 7–25. Leiden: Brill, 2007.
Speros Vryonis. The Decline of Medieval Hellenism in Asia Minor and the Process of Islamization from the Eleventh through the Fifteenth Century. Berkeley and London: University of California Press, 1971.
A.C.S. Peacock, Bruno de Nicola and Sara Nur Yıldız (eds), Islam and Christianity in Medieval Anatolia. London: Routledge, 2016.