After the colapase of the Great Mongol empire. the territories were devided in to a couple of smaller Khanates
Obviously, after the collapse of the Great Mongol empire, Golden Horde, Chaghatay Khanate, Ilkhanate, and Yuan dynasty emerged.
Khanates of ex-Golden Horde you are talking about emerged later. Most of the dates are arguable.
And please do not try to give these khanates an "ethnic" flavor. For many centuries "Tatar", "Noghay", "Ozbek", and "Qazaq" were simply political terms, not ethnic ones. They became fully ethnic later.
These are the major khanates of ex-Golden Horde (Desht-i Qipchaq):
So, Qazaq and Uzbek are the same people talking in different dialects (Qazaq in Kypchak, Uzbek in Qarluq) of Turkic.
No. Both Qazaqs and Sheybanid Ozbeks are Qipchaq-speaking people.
In the 14-15th centuries "Ozbek" was a generic name for the nomad of Desht-i-Qipchaq.
In mid-15th century, Ozbeks of Desht-i-Qipchaq split into two factions: supporters of Urusid dynasty and supporters of Sheybanid dynasty.
Eventually, the latter ones were driven out to Mawarannahr. Ozbeks who supported Urusid dynasty remained in Desht-i Qipchaq and later became known as "Qazaqs". So, all of the Qazaqs were once called Ozbeks.
As for the Sheybanid Ozbeks, they migrated to Mawarannahr (today's Ozbekistan) and mixed with the local Iranian and Turko-Mongolic tribes. Today, all of these groups are known as "Ozbeks", but in fact the real Qipchaq-speaking Sheybanid Ozbeks are the minorty among the modern Ozbeks.
The Uzbek nation has been formed of three ethnic layers: the urban population, the descendents of the pre-Shaibanid Turkic tribes, and the descendents of the Shaibanid Uzbek tribes.
The first layer is comprised of the indigenous Iranian population of the area who were subsequently turkified by the influx of Turkic tribes into the area in the first several centuries of our era. The presence of this element in the ethnic makeup of the Uzbeks has resulted in them actually being culturally closer to the Tajiks, who are of Persian stock, than to the other Turkic peoples of Central Asia. Furthermore, physiologically, both the Tajiks and the Uzbeks are very similar in appearance.
The second layer of the Uzbek nation is comprised of those Mongol-Turkic peoples who lived in Transoxiana prior to the Shaibanid invasion in the fifteenth century.
The final layer is that of the Shaibanid Uzbeks.
The result of such an ethnic intermixture is that one can see a blending of both Persian (Caucasoid) and Asiatic (Mongoloid) features in most Uzbeks, although the tendency towards one or the other is more pronounced in certain areas.
By this time, the Uzbeks had begun to mix with the previous inhabitants of the area, both Persian and Turkic, and many of these non-Uzbeks living in the area began to be called by the name "Uzbek" also.
...further distinction was made between the nomadic and sedentary population of what is now Uzbekistan. The latter were called 'Sarts' by the Russians, a group which represented a "cross between the aboriginal Tadzhik (Iranian) inhabitants and their Uzbek (Turkic) conquerors,"27 whereas the former were referred to as Taze (pure) Uzbeks. The subsequent grouping together of all these elements into an Uzbek nation is primarily a phenomenon of the Soviet period. In general, it has been a successful strategy, and most inhabitants of Uzbekistan today identify themselves as Uzbeks, although there are still some who maintain their tribal identity, referring to themselves by such names as Kypchaks, Mangyts, Karluks or Lokays.
I didn't mention White Horde and Blue Horde because they're still controversial: their geography, dates, rulers, and whether they were the same thing.
For example, Urus khan is frequently called a khan of the White Horde, but his people were called "Ozbeks", and his state was basically the same as the later state that is usually called Ozbek khanate of Sheybanids. But Deshti Ozbeks got their name during Urus khan, and, therefore, counting the history of Ozbek khanate from the later Sheybanid dynasty is wrong.