yes, the symbol was used by Turks in the past and by other groups but the Ottoman adoption of it is due to two factors one a legend about Osman the First's dream, and because it was the symbol of Rome and the Ottomans thought of themselves as the continuation of Rome. the Ottomans were forcing Caesar not Constantinople to stop calling themselves Roman for a while before they conquered Istanbul.
Culture is something that is tied down to one group it is something that evolves over time from different cultures coming together and exchanging ideas, for example, Greek culture gets most of its roots from the middle east.
The Crescent-star wasn't an Islamic symbol but there was a legend has it that Osman I put a crescent on the top of a mosque after he had a dream that he believed to be a vision. The Crescent-star is associated with the Islamic world mostly because it was used by the Ottomans.
Only Fatih Sultan Mehmet considered himself a Ceasar. Being called Caesar was a political title as he conquered Constantinople he wanted to conquer Rome next.
Ottomana called the0mselves Kings, Emperors, Caliphs, Khans and other titles to elvate their empire.
Even the titles Amir Ul Mumineen were used the Ottomans try to legitmatise themselves as being descended from the Quraish. It wasnt taken seriously even by themselves because all those titles were for political legitamacy.
In many ways the Ottomans did not really consider themselves Romans. It was just another title to add alongside being the Khan.
Even the Russian Tsars used Khans in their titles. Did not really make them Mongols or Gokturks.
