If you've got a bilingual society I can imagine it makes life easier if both languages use the same alphabet, particularly in the days of mechanical typewriters. But think too about road signs, for example, which could be written in the local language but still mostly comprehensible for Russian speakers who don't know the local language if it's the same alphabet (though there's still the classic problem of "Why are half the streets round here called Einbahnstraße?"). Learning to read a new alphabet at speed is surprisingly difficult. I'm surprised by how hard it is for me, anyway.
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.
That doesn't seem correct to me. Different languages require different orthographies because they have different phonetics. Compare the Latin script when used for English to German to Vietnamese to Turkish. Same thing with the Arabic script with Persian and Urdu. When tailored orthographies aren't used, the written representation becomes a poor representation of speech. Though even then it might not have been a perfect fit in the first place, requiring digraphs or diacritics. Or the spoken language can diverge.
For example, Turkic languages typically feature front/back vowel harmony where stops are velar before front vowels and uvular before back vowels. This is represented differently in the Stalin-era Cyrillic orthographies for Karachay Balkar (к/къ), Kazakh (к/қ) and Tatar (just к with the following vowel letter showing the distinction). Bashkir neighbors both Tatar and Kazakh and is mutually intelligible with them, and it doesn’t differ at all in this feature, but its orthography was given к/ҡ just to make it different from the other two.