I was looking at the [map] (https://upload.wikimedia.org/wikipedia/commons/e/e5/Map-TurksoyMembers.png) of International Organisation of Turkic countries (Turksoy) which led to this question.
Also, why were Georgia and Armenia not influenced by the Turkic culture but Azerbaijan was considering how these 2 countries are situated in between Turkey and Azerbaijan?
"Turkic" refers to an ethno-linguistic group that originated and developed on the Eastern Eurasian steppes, it doesn't refer to the nation of Turkey; the earliest Turkic speaking peoples were found in the Xiongnu Confederacy, which existed in present day Northern China/Mongolia. Turkic migrations and invasions occurred throughout history over Central Asia, and they established a lasting presence. Modern-day Turkey is called so because the Ottoman Empire was founded by Turkic tribes migrating west into Anatolia, and thus Europeans called them "Turks".
Eventually, Turkic peoples intermingled and founded many empires in the Middle East/South Asia, like the Ghaznavid, Timurid, Mamluk, Ottoman, Safavid and Mughal empires -- which all shared Persian influence as well. However, it's still important to note that the Persian ethno-cultural group is Indo-European in origin, which is very distinct from the Turkic ethno-cultural group.
Many Central Asian countries are called Turkic because their ethno-linguistic groups are Turkic, despite their closer historical relations with Persia.
Well, the majority of the people in Central Asia, with the exception of Tajikistan, are Turkic in both ethnic makeup and in culture. Uzbeks, Kazakhs, Kirghiz, and Turkmen are all considered descendants from the ethnographic tree of the Turko-Mongol group. This means that they came with the Mongol invasions in the 13th century and absorbed elements of existing Turkic culture and tribal groups alongside their own Mongolian culture. Uzbeks for example are descendants of Mongols and Turkic nomads who were under the jurisdiction of the Khanate of Jochi.
The only exception to this group is the Turkmen and Uyghurs. They were ethnically Turkic in nature, existing during the period of the Gok Turks in the 5th to 6th centuries who were influenced by Mongolian culture and standards thereby adopting, as similar to the Uzbeks, an Turko-Mongol identity. The only difference between Turkmen and Uyghurs are that the Uyghurs are considerably more settled and not as nomadic as the other groupings.
Tajikistan is different in that the Tajiks were an Indo-European people who were primarily settled and had some form of agrarian culture. This group was generally where Turko-Mongolian Khanates drew their administrators and bureaucrats from. While they maybe influenced by some aspects of Turko-Mongolian culture they retained their Persian culture.
In regards to your second question, Azerbaijan was subject to multiple invasions of Turkmen from the period of the Seljuks in the 10th-11th centuries until the rise of the Safavids in the 16th century. These constant invasions, empires, Khanates and nomadic groups changed the make up of Azerbaijan to where it essentially changed their ethnic makeup.
One of the reasons why Georgia and Armenia retained their culture was that their own sense of culture was strongly established and that the Turkmen found that there was no need to actually move into their lands. Azerbaijan, compared to the other two, has a more suitable environment for nomadic life. Besides, the existence of Armenia and Georgia provided a fertile ground for raids on the basis of religion where the Khans could direct more rebellion prone nomadic groups towards to divert their energy towards rather than cause a rebellion within their territory.
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I think one thing isn't clear in your question, what do you mean by turkic culture? what else defines that other than just the language?
