Why really? Don’t they speak Turkish? What makes them different? Well, you’ve come to the right place.
To begin with, we have to start from 6th century. The age of chaos in Central Asia. Famine, population and climate changes (yeah, climate change) created a need to emigrate Altaic nations (the word “Altaic” is not used here to indicate Altaic Language Family hypothesis which is pretty dead today, the area is meant here) in all directions. For centuries, Altaic nations raided Trans-Caspian Steppes and Eastern Europe, establishing vast federations (European Huns and Sakha is meant here) back then, even wars happened between them and Roman Empire. But this time, raids would become a necessity, a struggle to survive. Some Turkic groups were already divided at that time like Sabir in the North Siberia and Oghuzes in the shores of Caspian Sea: This is the story of Oghuzes.
After the collapse of Gokturk Khanate in first half of eighth century, Oghuzes, one of many Turkic tribes and one of the biggest, have already been immigrated to west establishing several states like Oghuz Yabgu state and Kharakhanid Khanate. Main mass in these states are Turks, especially Oghuzes but they are not alone to be sure. First sparks of blending have happened, we could discuss why for hours but Turkic culture started to became a superior culture in South-West Central Asia. Many Indo-European cultures like Tocharians and Indo-Arians would became mixed with Turkic peoples, became assimilated under the culture, married them, their native language became Turkic after few generations, becoming more like them, Oghuz phenotype started to change but this change would take more centuries to come. By the year 1037, Tughril and Caghri established Seljuk Empire, causing Turkic mixation with Armenians and Persians, Oghuz phenotype became even more like West Asian, creating a very close look to modern Turkish nation. By the collapse of Great Seljuk and Anatolian Seljuk and establishment of Ottoman Empire, even Balkan and Greek phenotype affected the phenotype, creating today’s Turkish look.
But Turkic rule of Persia with Safavids for ages affected Azeri people even much more than Turkish. And under Romanian and Slavic rule, Gagauzes became more “European”. Even though, Oghuzes still speak a Turkic language and can mutually understand each other. They preserve the main elements of Turkic culture. Even after all these years. And they are persistent calling themselves Turk more than any other Turkic nations.