That was the original idea behind .name. You buy firstname.lastname.name. When it wasn’t a huge success they got bored and it reverted to a regular tld.
Interesting, but also a bit worrisome idea. I wouldn't be comfortable knowing that someone else can buy the adjacent subdomain anotherfirstname.mylastname.name and be associated with me in the eyes of anyone who doesn't know how that TLD works (which would be almost everyone).