I don't think there are many people in my life that I've called by just 1 name.
I'd say it depends on the person using the name and the person being named. And how you define "names" in the first place.
Personally, I almost
never call people by different names - except possibly over the span of
years. ie, someone you know as Andy as a kid might prefer to be called Andrew as an adult because it sounds more mature to them, or you might go from calling someone "Mr. Smith" to "Robert" to "Bob" as your relationship with them changes. Over shorter periods of time (like in the time period spanning most VNs), I'd generally have one name I use and stick to it all the time.
If anything, context shifts usually matter for me more for "titles" than names - for instance, if I'm talking to a friend I might say "I heard your mom went to Vegas", but when talking to someone else I might say "I heard Bob's mother went to Vegas." Or if I'm talking to someone who knows Bob's mother by her name but doesn't know Bob very well, I might say "I heard Alice went to Vegas." Or conversely, if speaking to someone who doesn't know either of them, I might just say "I heard my friend's mother went to Vegas." That's not really different names - Alice is always Alice - as much as it is different ways to describe someone.
But in a story I don't think I'd ever be bothered (or even notice, honestly) if I only ever refer to a character by a single name even if I
did tend to constantly call people by different names. Most stories do that anyway (even if you can't rename characters) simply to reduce confusion on the part of the audience (the same reason why most stories don't have three characters named Steve).
Not only that you remove some power of the writers. You can't ever use a variant of their name as a term of affection from for instance their family or the mc.
The trade-off is that you're empowering the player/reader. Which is something I almost always prefer when you're talking about a narrative choice game/interactive story.
There are multiple reasons why a player might want to change a name. Maybe changing names to match people they know personally allows them to better immerse in a story. Or using names they like helps them care more about a given character. Or maybe they just think the original author-intended names are stupid and can't stand them (honestly, when I was growing up, I'd have enjoyed Final Fantasy IV a
lot less if I'd been forced to play as "Cecil" instead of getting to change my name).
And honestly, allowing the player to rename characters doesn't make terms of affection impossible, it just means nicknames can't be derived from the original name. So Barbara can still call her daughter Alice "Cupcake" and Alice can still call her mom "mom" even if you've renamed them both to Anastasia and Zoey. Sure, "Charles" can't be Chuck or Chaz or Charlie or C-Dawg if you can change his name, but he can still be "bro" or "honey" or "Ace" to other people based on the context of their relationship to him.
The main problem with being allowed to rename characters is when you're making a game with voiced characters (because other characters will never be able to call you by your name). But there are ways around that as well: games like Fallout 3 and New Vegas just have people refer to you by a "title", while Fallout 4 has the robot that can literally recognize hundreds of names and use them, or will default to sir/ma'am if you pick a name it doesn't know. Meanwhile, Dragon Age 2 just has everyone call you by your last name, and Final Fantasy X just had people avoid using any name at all and only directly addressing you (so they never needed to refer to you in the third person to someone else).
But
very few VNs are voiced, so it's not really a problem that applies.