So they wanted to be called Romans, that's why they said to people who called themselves Romans that they are not Romans, anymore
Wow, a lot to unpack here.
True only in the most general sense. The Ottoman state was composed of multiple "nations" (millet) identified by their religion and language. Turkish Muslims and Greek Orthodox Christians were the main (or core) "millets." In a sense, the Ottoman Empire was a Turkish-Greek empire with today's terminology. The demographics of Roman lands didn't change immediately just because the head guy is now a Muslim, and Byzantines vs Ottomans was a political struggle rather than an ethnic or religious one.
Not everything changed with respect to laws and customs, either. millets (except Muslims) had autonomy and were ruled by their respective religious leaders. So subjects of the Greek Orthodox Church were under the rule of their patriarch. They had their own laws, lands, etc. This meant that they continued some laws and customs from the Byzantine Empire.
Ottomans took over a lot of laws and state instutitions from the Byzantines. They started keeping state records, history, etc. One can suggest that they became a proper "state", which they went on and named "the Grand State." It is such a pivotal point that from then on we call it the "Ottoman Empire." It was much more of a continuation of the Byzantines than a continuation of the Ottoman Beylik. But at the end of the day it was neither.
It changed a lot