I don't mean owning in the sense of having some kind of copyright. I mean that some anime characters have become so closely associated with their given names that you can just refer to them by name, without mentioning family surname or series.
This is not a matter of your favorite character, or the character you think ought to own the name. Think of it rather as walking into a room full of weebs of all varieties, throwing a name at random, and asking who they think you mean. Or mentioning the character in a general discussion thread here (as opposed to a series forum).
This is certainly related to popularity, but not always. "Hitagi", for example, is a character everyone knows even if they haven't watched the Monogatari series, but most people use "Senjougahara"; I'm sure many don't even remember her given name. Also, the name itself is very rare; there's no merit in owning it if you're the only one.
An extreme case, by way of example: Haruhi Suzumiya started airing on Apr 3 2006. Almost at the same time (Apr 5 2006) Ouran Koukou Host Club started, with Haruhi Fujioka as protagonist. Host Club is by no means an unknown or forgotten series, but there's no denying that "Haruhi" evokes first and foremost H. Suzumiya. You have to dig very deep in character databases to find any other Haruhis, especially after 2006 - as if creators had actually stopped using the name except for very minor characters, since the associations are too strong.
Another name that comes immediately to mind is "Asuka". This one is much more common, and new Asukas pop up all the time, but again, if you use the name alone, people will assume you mean the bitch from NGE. If you mean another bitch, you need to say "Asuka from Sounan Desu ka" or "Asuka from Hibike Euphonium". Similarly, "Madoka" is definitely the magical girl, but not to the point that they have stopped giving the name to new characters.
This is also something that nacessarily changes over time. When I was a kid "Misa" was absolutely taken by the Macross girl. These days it's no longer the case except among die hard fans.
So, which names are more or less permanently "taken"?
Here's a list of the 20 most common given names that I found somewhere, as a memory aid. But surely you can think of others:
I Believe it has to do with something about popularity. Lets say, In the Example of Haruhi, who would have the most popular name, or at least the most associated name? Is it the one that literally has the name "Haruhi Suzumiya" as its anime's title? of course would.
its all about how the anime likes to refer to them as. Hitagi isn't directly connected to Monogatari; This is primordially due to this name being rarely used, Because most of the characters in the show talk to her and name her by her last name, senjougahara, especially the main character, araragi. Basically, If a Popular anime really sells a name and use it frequently, the characters name might be associated with this anime; of course this doesn't happen all the time, but at least myself has associated a name to a character sometimes.