Why are Turkish people and other Turkik people considered of Mongolic race if their language is not in the Mongolic language family?

Your very question has no answer since it is factually wrong — most of population of modern Turkey belongs to Europoid race (more specifically — to Mediterranean, I guess).

However the answer to any question in a form of “Why X belong to Y race, but speak Z language” is because race is biology and language is culture.

I hope you are not very surprised by the fact that both Europoids of Spain and (some) Americanoids of Central America speak Spanish, or that both Europoids of Arabia and Negroids of of modern Sudan speak Arabic. Reasons to that are only few centuries ago from now so they are well-known: colonization and islamization respectively.

Even though turkization of Lesser Asia (≈ modern Turkey) dates back to IX century — pretty Historic age, the fact that language of Turks is akin to languages of Mongoloids of Central Asia might be surprising for someone. What to say about events that belongs to Prehistory...

I am writing this because I believe the Wikipedia article on Turko-Mongol tradition does not really help readers understand the concept, which gave rise to this question (my assumption).

Also, (perhaps) this question stems from a confusion between genetics (biology) and genealogical relationship (linguistics). If that is the case, this article on language family might help clarify: genetic relationship (linguistics).

Having said this, the answer given by T.E.D. is correct insofar that the proposed Altaic language family is now discredited because, if the hypohtesis is true, earlier forms of Turkic and Mongolic language should be more related (linguistically, not biologically) than later forms.

Instead, comparative linguistics has shown that, in fact, "the earliest written records of Mongolic and Turkic languages shows fewer similarities rather than more, which suggests that they do not share a common ancestor, but rather have become more similar through language contact and areal effects. Because of this, most modern linguists do not accept the Altaic family". Source

Again, back to T.E.D.'s answer, on his statement "the similarities are more due to long proximity than heritage" -- this is areal effects.

So, in fact, the present day similarities of Turkic and Mongolic languages is due to a long period of contact and was expedited by Mongolian expansionist policy, which gave rise to Turko-Mongol tradition (14th century).

To end, here's a paragraph from Turko-Mongol Relations (chapter twenty-one of The Mongolic Languages, Routledge, 2003):

A more plausible explanation (for the established Turko-Mongol tradition) is therefore offered by the assumption of a network of linguistic contacts, which have united the ‘Altaic’ languages since ancient times up to the present day. External contacts with Uralic and Indo-European suggest that Proto-Turkic was once the westernmost member of this network. On the basis of historical information it may be concluded that the period of the most intensive early contact between Turkic and Mongolic coincided with the appearance of the protohistorical ethnopolitical entities of Xiongnu and Donghu in the regions north of China (Mongolia and Manchuria).

Also, see Northeast-Asian sprachbund.

First, Turkish people are not in general considered of Mongoloid race, neither are Turkic-speaking Azeris. These two peoples are usually classified to belong to Europoid race with maybe only admixture of Mongoloids.

Second, there is a lot of Turkic-speaking peoples that are indeed classified Mongoloid: Kazakhs, Kyrgyz, Uzbek, Turkmen.

The reason is that Mongoloid race was named for Mongols who were a typical representation of the race, but includes also a lot of other peoples with totally unrelated languages: Sami (original), Nentsi, Mansi, Yakuts, Chukchi, Japanese, Chinese, Korean, Thai, Indonesians, Eskimo, Aleuts, Native Americans, etc.

Mongoloid race has multiple defining characteristics, but one typical feature is the existence of epicantus over the eyes.

There's no scientific definition of "race", so that part of the question is unanswerable.

Linguists do try to classify languages though, since that can be done (mostly) objectively, based on similarity of grammar and words. One popular theory has been Turkic languages and Mongolic languages are part of a larger family of languages, named Altaic.

Today this is no longer a universally accepted theory, with the alternative being that the similarities are more due to long proximity than heritage. But Altaic was the consensus accepted theory until about 15-20 years ago (which is practically yesterday for oldsters like me). So it can be found in many textbooks and references floating around today.