Would I get in legal trouble in the US or UK for starting a company called Elon and selling 'musk' cologne? I know that College Humor made a joke about this a few years ago, but this is an actual business idea. I have already bought the domain name elons-musk.com.
Your question actually concerns a trademark issue and not a copyright issue, because it involves the name used in association with the sale of a specific product or product line.
In isolation, "musk" is a generic term for certain fragrances and cannot be trademarked.
There is also U.S. case law (one involved a lingere shop called "Victor's Secret" run by a fellow named Victor from birth, if I recall the facts correctly, the held that someone can use their own personal name not adopted for business purposes in the name of a business they run personally, even in the fact of a strong, nationally known trademark in the same subject matter area that could be confused with the famous trademark.
So, under U.S. law, this would probably be allowed, although it might draw a challenge. I don't know the parallel caselaw in the U.K. (FYI, Victoria's Secret is an Ohio company started with a small storefront in London which exists just to give it an exotic address for marketing purposes until it got big.)