It is spelled just so, contains just these words in just this order, and it will always be that way. The reader can then count on finding that exact name in lists and directories, and can pass the name on to others with confidence. Names have a special precision and significance that common nouns don't have. Let's not pollute that significance by capitalizing everything that won't run away. Don't capitalize job titles, technical terms, or spelled-out acronyms, unless they are names.
Uniqueness is irrelevant.
If someone tells you it's a name, check it out. When in doubt, don't cap. The reverse is not true. Try to gently educate people who measure their own importance by the number of capital letters attached to their work. It may help to point out that unneeded capitalization is a literary device for indicating satire or conveying pomposity.