Roughly, latinization was a Lenin-era policy. However, when Stalin came to power he had different goals and philosophy than the preceding regime of Lenin. For one, Stalin abandoned the idea of spreading world revolution in favour of "socialism in one country", which in large part was about keeping the USSR (and, as its satellite, Mongolia) walled off from the outside world for the sake of tighter control.
For speakers of the USSR’s Turkic languages (Kazakh, Tatar, Crimean Tatar, etc.), latinization was therefore seen as harmful, because these languages are often fairly mutually intelligible with Turkish of Turkey, and a common-ish Latin alphabet would mean these peoples would be susceptible to anti-Soviet ideas coming from Turkey. Therefore, the Stalin-era language authorities not only gave them a new Cyrillic orthography, they gave each language a different Cyrillic orthography than the rest to fragment them even further.
In the case of Mongolian, Cyrillic helped better separate Mongolian of Mongolia from the mutually intelligible dialect of Mongolian spoken across the border in China.
For speakers of the USSR’s Turkic languages (Kazakh, Tatar, Crimean Tatar, etc.), latinization was therefore seen as harmful, because these languages are often fairly mutually intelligible with Turkish of Turkey, and a common-ish Latin alphabet would mean these peoples would be susceptible to anti-Soviet ideas coming from Turkey. Therefore, the Stalin-era language authorities not only gave them a new Cyrillic orthography, they gave each language a different Cyrillic orthography than the rest to fragment them even further.
In the case of Mongolian, Cyrillic helped better separate Mongolian of Mongolia from the mutually intelligible dialect of Mongolian spoken across the border in China.