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.
Speaking as someone whose mom didn't change their name when marrying my dad, with a sister who didn't change her name when marrying my brother in law, with a wife who also didn't change her name when she married me, I think this problem is overblown. I have yet to encounter any actual issues with this.
Sometimes people will assume we aren't married and/or divorced, and people will often call me by my wife's last name and vice versa, but it has never caused any actual problem. Never had any system that assumes we have the same last name. So many people live in blended families anyway, that very few systems/people make these assumptions any more.
People getting the issues live in different systems and/or have different needs, and it also changes with our world getting more digital. One part that doesn't much depend on locality this days would be international travel and money.
For international travel, small kids having a different name is surprisingly painful and can get you stuck in an office for hours until it's somewhat clear you're not kidnapping them (proving you're a parent not being enough). Depending on how it goes your plane could be gone by that time.
Money is the same, there;s a lot less check if you send to yourself or family than to a random stranger. Having a different name can mean your transfer getting stuck for days of back and forth.
Then again, if you're just staying in your town never dealing with anything outside of it, you might never have to think about your name in your whole life.
I never experienced any of what you say as a child. We travelled internationally a number of times, never had anyone tell us she wasn't my mom or anything.
I am not sure what you mean about the money thing. My mom was on my accounts when I was a kid (with different last names) so she could send me money. As an adult, I can't see how sending money would be an issue. My mom and I transfer money to each other fairly often still ($70k recently, went through fine).
I did not stay in my home town. Not sure how they would be relevant.
Again, most people would assume (if they assumed anything) that my parents were divorced, which is incredibly common. Half my friends had different last names from their siblings and/or parent. Blended families are incredibly common.
I am now a dad of two. They have my last name, while my wife (their mom) has a different name. Again, never a problem at doctors or school or anything. They always make you fill out your full name and relationship. Again, super common to have different last names here im California.