Although probably a majority of interested linguists believe in Altaic, it is not established beyond all doubt. The
numerals, for example, show no great similarity. The resemblances might be because they are all part of a
Sprachbund, an area where large-scale borrowings have propagated across family boundaries.
It consists of three families: first, the Turkic family (including Turkish, Uzbek, Tatar, Kyrgyz, Kazakh, Azeri, Yakut); second Mongolian; and third the Tungusic family (principally Manchu), but the branches might not all be related as directly as that.
It is sometimes connected with the Uralic languages (Finnish, Hungarian, et al.) in a family called Ural-Altaic, but that is much less certain. Some linguists also want to connect it to two or three isolates in the area, namely Japanese, Korean, and Ainu. These also share a lot of superficial similarities, and it is tempting to want to see them as related. The most conservative hypothesis is that Turkic, Tungusic, Mongolian, Uralic, Korean, Japanese, and Ainu are all separate families; the boldest conjecture is that they all form part of an even wider Nostratic or Eurasiatic super-family.
What the Altaic languages share includes vowel harmony and agglutination. The personal pronouns seem similar, but this could be coincidence in such short elements.