So Russian families who move to America have a choice - either deal with people and systems who assume that married couples, and parents/children all have the same last name and hit roadblocks when that expectation does not match reality, or change one partner's last name to match the other's.
But that second option has problems too, because that name change doesn't retroactively apply in Russia - so now you might have American documents that say you're a Elena Kuznetsov, but your Russian documents say that you're Elena Kuznetsova - so any legal dealings that involve the two countries (like, say, traveling) become significantly more complicated because you need to prove that the two names actually point to the same person.
At least middle names aren't a big issue - patronymics mean something in Russia, but here in America it's just a string you pop into the "middle name" field, and maybe you get asked what it means, and get to teach someone what patronymic means.
(It's a more general question, too, is John Juan when he's in Mexico?)
Although it's not quite that simple because the original version of Cyrillic actually has a bunch of extraneous letters that are there solely to represent the distinctions in Greek; in some cases, distinctions that were themselves historical in Greek by that time even. For example, three letters for /i/: I (corresponding to iota), И (corresponding to eta), and Ѵ (corresponding to upsilon), all of which were already pronounced the same in contemporary Greek, and this carried over to Slavic languages as well.
In other cases the distinction became nativized though. E.g. Greek theta, already pronounced as /θ/ in Greek, became the Cyrillic Ѳ - but the closest they could get to pronounce it was [f], and so it came to have the same meaning as Ф, and eventually Ѳ was just dropped as unneeded. Thus e.g. transliteration of Matthew is Матфей, and a bunch of other words where most European languages have "t" or "th" sounds have "ф" in East Slavic languages: e.g. "arithmetic" is "арифметика". But then some words were borrowed into Russian from Latin as well, or from other languages that borrowed them via Latin, and so sometimes theta became /t/: "mathematics" - "математика".