I think names tend to just become a part of who we are and how we think of ourselves. Think of your first name for instance, though some I'm sure would happily change their first name, it still becomes a part of that person. When we meet a new person or even a new pet, the first things we take in are their appearance and their name. I think it is a natural identification process going on there. Later as we get to know them better, we start associating their personality with their name. I think that is why sometimes people will develop a dislike to a certain name too, if you grew up or spent a lot of time around someone who's personality was cruel and mean, and you didn't know anyone else by that name, I'd be willing to bet you'd carry some association with the name. How many parents would want their child's name to be Hitler? You see where I'm coming from a bit here? It's hard to put into words.
I know with my dogs, when I hear or say their names, they aren't just a name, I have emotions and feelings that are elicited by seeing and hearing their names, associations.
Perhaps if we got up from day to day and said "I'm going to be Amy today" or "I'll be Shirley this week," names wouldn't have such a sense of identity, but it is the very first way we identify ourselves to the world whether it be greeting someone informally or putting it on legal documents. I think after awhile, it just becomes a part of who we are.
I do think in some cultures and in the past, names had more actual meaning, even surnames. Someone who's family livelihood was tailoring might be the Weaver family and so on. A house name from medieval Europe was a complete definition of that person's place in society. I do think last names definitely have less definition to them now than they did at one time.
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