Never forget how when Italy first introduced Romano to Germany in the English dub, he did so by saying, "Since we've always been governed separately, he had to spend more time with our big brother Spain, so he's become kind of a dick!"
Italy, in canon, insulted his older brother to his face while introducing him to someone. And with a smile, I might add.
Now, a popular theory of why nations have human names is that the nations use them when interacting with humans. My personal headcanon is the opposite: it is for each other.
One of the things I found while researching history is that nations often go through many different name changes. Just look at China. He went from Xia to Shang to Zhou to Qin to Han... ect. Imagine how hard that would be to keep track of?
India: "Hello, Qin. It is good to see you again!"
China: "Oh, I'm not Qin anymore, I'm Han."
India: "Wait, you've changed your name again?! How many civil wars are you going to have?"
This isn't even getting into the fact that different languages call countries different things. For example, the citizens of China do not call their nation "China," they call it "Zhōngguó." The name "China" is the English version of the native name Zhōngguó. As you can see, even if a nation's name remains the same, it can be totally different depending on the languages
So, I personally headcanon that human names are something nations can call each other that will always remain consistent, even through language barriers and even if the name of their nation changes.
China will always be Wang Yao (or Wong Yue), even when his nation is called Song or Zhou or Qin. If nations are talking to each other, and one doesn't know who "Zhōngguó" is, the other can just say, "I'm talking about Wang Yao. Zhōngguó is the new name he is going by now."
These human names are universal and, for the most part, don't go through nearly as dramatic of changes as their nation names when it comes to language variation. For example, for English speakers, the People's Republic is "China." To Chinese speakers, it is "Zhōngguó." Meanwhile, "Wang Yao" remains "Wang Yao" across the languages, it just varies in pronunciation between English, Mandarin, and Cantonese.
Basically, what I'm trying to say is that nations having human names is both a simplicity thing and a matter of convenience. A singular name that will never, or barely, change no matter how much time passes is very useful for immortals whose other names are constantly changing.
Of course, besides simplicities sake, I think there could be other reasons too.
For example, I think it could also be an exercise in autonomy. They can't control what their nation is called, but they sure can pick their human name.
They do this to pick human names that they think reflect who they believe they are, or perhaps because they simply think a name sounds cool and they want to be called it. Regardless, they get a choice in this name while they get no say in their nation one. This isn't to say they hate their nation names, most love them! But there is just something different about a name you chose for yourself.
You can see with China, for example, that his human name is actually very reflective of him. The first character in his name "王/Wang" meaning "king," and the three lines in it represent heaven, earth, and people, which represents the king's rule over China. The last character "耀/Yao" means "shining" or "glory," which feels like the type of name China would call himself. He wants to let everyone know he is heaven's gift to the world, lol.
Another reason they might have human names is that it makes them feel more... human.
They have a sort of twisted existence. They exist because of humans and their continued existence is entirely tied to humans, but they also can never really be connected to humans. It is canon that, if a nation spends too long with a human, their immortality will rub off on the human and drive them insane as the human mind can't comprehend immortality. This isn't even mentioning how short human lives are compared to them. See: Davie.
So, nations live in this limbo of existing because of their connection to humans but also never being able to truly connect with the very humans that make up their existence. Because of this, their use of a human name makes them feel artificially closer to humans, even when they know they aren't.
Now, do I think they would use human names all the time? Do I think it would be rare? How much would they use their human names compared to their nation ones?
This sounds like a cop-out, but I think it totally depends.
I believe it would totally depend on the culture and also the relationship between the two nations. In general though, I think that the closer nations are together, the more likely they are to use human names. Or, if the nation doesn't stress over the use of names as a sign of respect and are quite laid back, they might also be fine with anyone using their human name.
Also, culture plays a part. Americans, for example, are much more laid back about names than compared to East Asia. Ergo, I think America would be totally cool with people using his human name. Heck, he might even insist as proof he and that person are "buddies."
On the other hand, some nations might absolutely expect to be referred to as their nation name only. Yeah, they have a human name, but that is reserved only for the people most precious to them. If they are a conqueror, then they expect the conquered to refer to them with a nation name. They are a nation, dammit! Not some measly human, so don't use the measly human name. Use their great nation name, instead! They didn't become the greatest empire in history to not be referred to with the name of this great nation!
I know I have been using China as an example of why it is good to have human names, but I personally headcanon he was actually a pretty big stickler of making others refer to him as his nation name only, especially his vassal states. Yeah, knowing what to call him got confusing, so what? He is worth the effort.
(Though, I do think he relaxed on it as he got older, especially after the communist revolution.)
On the other hand, nations he saw as equals and/or was close to, he would allow them to use his human name. Like Ancient Rome, Russia, and India, for example.
One last thing I want to add is that I think the biggest reason nations all have human names is just plain, ol' peer pressure.
It sounds a bit silly but, I think when a bunch of nations started using human names, other nations saw this and thought, "Crap, everyone is getting a human name! Do I need one too? Will I fall behind if I don't adapt to this new trend?!"
Eventually, it got to the point where all nations had human names and all the nations that came after them just saw the universal usage of human names and assumed that they had to pick one, that it was just part of being a nation. Now, by modern times, very few nations even know there was a point where human names weren't universal. Having human names even got to the point where nations were bestowing these names on younger nations the way a human parent would name their child (like my headcanon of England giving America his name "Alfred" after Alfred the Great).
Honestly, this is a really long headcanon for something that probably isn't anything more than an easter egg in canon. But, tldr: human names make it easier for nations to speak with each other when their nation names are always changing.
Anyways, here's a fabulous China gif since I talked about him so much. Thanks for reading!
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