I think both ways of doing it are valid, and your examples already show that this is the case. Both can be interesting.
But you are asking about advantages/disadvantages. This is a much more interesting question than "which one should I do", so I'll try to live up to it. I will say though that I slightly prefer keeping the formalities because I think it's just another tool for world building and it would be wasteful not to use it. So my answer might be a bit biased.
Advantages of keeping formalities between friends
If done right, these formalities can be a glimpse into the fantasy world that we are inhabiting. For example, people from different cultures might have different ways of addressing one another. It can therefore be used as a tool for characterization, but also as something that can cause conflict. Imagine you have two close friends, and in a heated argument, the "lower" one stops addressing his friend as "my prince", but simply calls him "Henry". That might actually increase the tension. These social rules are exactly one of the reasons people like reading fantasy - they're a part of the world building. We do not want to have everything the way it is in our own world. (I would also say that if you do it, then think of your own formalities that you can use, and make them different for different cultures, and so on. If you write a fantasy novel but everyone talks like they did in medieval England, it's boring and very been-there-done-that.)
After a while, it might grow tiresome for the reader, but I think that can be avoided with good writing. It should not really matter whether someone calls his friend "Jonah", "sir" or "my liege". The bigger problem might be that the reader mistakes the formalities that are very normal and common in this fantasy setting as something that is toxic within the friendship. You know, if you always maintain the hierarchy between two friends, it kind of seems like they are not friends at all! So you have to be very careful how you use it, because you do not want to give the wrong impression. You want to show that, on a friendship level, they are equals. If you have a friendship between a prince and a commoner, it might be necessary to show early on that the prince will literally risk his life or his good standing to save his commoner friend, or something like that, to make clear how true this friendship is for the two of them.
All in all, I'd say misrepresenting the friendship fundamentally is the biggest risk you are facing here, at least if you are good enough as a writer to keep the formalities from feeling boring or annoying.
Advantages of losing the formalities between friends
It's simpler for you and for the reader. It also quickly shows intimacy and triggers a positive reaction from the reader, because we tend to dislike upper class people who care about their titles. It might also give you more ways to play around with the dialogue, because if formalities are not a thing for these two, then it is also likely that nicknames, rude jokes and so on will come into play. It lightens up the dialogue considerably, basically.
Readers might not feel drawn into the world, because there's a dissonance between this "modern" type of friendship and what they expect from a fantasy setting like the one you describe in the book. You see this very often with people who argue that "in such a setting, people would never do X" and so on. All the explanations for why this friendship is so special that they call each other by their first names etc. might ring hollow, like a lazy excuse for the author to fit modern sensibilities into a setting that wouldn't normally allow for them. You know, just like when characters in a medieval fantasy setting suddenly demand equal rights for men and women or something like this. You better be a really good world builder to explain why this happens in this world apparently a thousand years before it "should" happen.
I think your second example of the guy who called his king by a nickname works because it is a true story. There are some stories that only work if we know that there is no author who specifically made the story the way he wanted to. If you want to have that kind of relationship, you cannot introduce it as an aside and then focus on your fantasy adventure - the story itself would have to focus heavily on how this special friendship came to be, and you would have to invest a lot of time to make it believable. Your example took the "shortcut" of being true. You cannot do that.
That said, I think your first example is also a good one that shows how weird it can seem at times to keep formalities in a friendship. I think we have all been there that we thought Samwise should just stop calling Frodo "Master". We do not want to read about a "friendship" that is actually a one-sided servitude without anyone talking about how unhealthy that is. (If I remember correctly, Frodo does comment on it, though, and does not especially like the way Sam treats him, so LotR has that covered.)
I guess in either case, no matter what you choose, you have to invest some time in justifying it to the reader. But I think the reader is technically on your side - we do want this type of friendship to work, but we also want a good explanation for it.