The modern Turkish identity was constructed in the 19th and 20th century under nationalist leadership within the Ottoman Empire and Turkey. Before then their really wasn't a fleshed out Turkish identity, and some can argue that there isn't one today, Muslims in the Balkans for example were called Turks for years even though they were Slavs. Or for many years anything coming out of Central Asia was simply referred to as Tartars.

However, that doesn't make your question any less valid, the Ottoman and Seljuks had a significantly different culture and identity to other central Asian Turkic groups. Much of this has to do with migration and intermingling with other native populations. The Oghuz Turks migrated south though Persia, and had strong Persian influences. The Seljuk empire is considered a combination of a Turkic and Persian empire. It's literature and its elite all spoke Persian. It's art, architecture, was all done in the style of the Persians. The Ottoman Empire, descendants of the Seljuks and the Oghuz Turks was heavily influenced by the Byzantium's, and the Arabs. Many historians refer to the the Ottoman Empire as being a continuation of the Byzantium empire, having adopted much of its court and cultural systems. This is the background that would shape the Turkish identity. There is of course a rather large cultural transformation in the 19th century, were an actual Turkish identity is created (as opposed to a religious one), but it was built of this history.