The name I live by in the regular world is certainly not removed or wrong to me but it is not "my name". It is not to say I am not aptly named or even not named in an ironic sense that would only play out regularly in a major book as part of my character arch and plot element, or that I disown or even dislike my name, rather it is that I simply do not like names as people do. I, of course, as a beast have a name - something I am recognized and called by - yet I am not confident the names animals have for one another translate so literally or cleanly to any human language because of the way animals communicate, felids included. So a name to me is simply what people call me, right or wrong. I have learned to acknowledge it despite it always feeling distant, a phenomena of what I would phrase as, "What I am correctly called by people but do not call myself."
Which leads into the fact that I place little stake on claiming a name for myself. I cannot give word to my true name but if I were to somehow hear what it is I feel about myself, as abstract a concept as that is, I would recognize it immediately. So it is neither here nor there in the end, which leads to me taking on "names", really descriptors, of what I am. I am "red in tooth" because I am a sabertooth cat, it is a direct appeal to what I naturally do and am as my true self. It takes little imagination why that phrase is descriptive and accurate at the same time and bases itself slightly in some very primitive language; Proto-Indo European, for example, has the root of "lynx" in some forms and iterations as "lyeuks" or similar spelling, and it translates to "bright eyes" or "shining eyes" - an obvious explanation to just a quality of the animal. This type of descriptive language is more accurate to me and more or less supported my habit of being largely "unnamed".