The point, Ladyleftfieldlover, is that Lilibet is not the Queen's name. It is a pet name which she was called as a child by a few very close relatives, because she couldn't say "Elizabeth" when she was small, but said it "Lilibet". When she was older, it was only her husband who used this pet name It is more personal than her given name, and has more emotional vibes around it.
Use your imagination. Suppose your name were Margaret, but you couldn't say that, so (charmingly) as a child you called yourself Migsy, and when you married, your husband called you that too, as a pet name. Then, later in life, one of your family chose to be nasty about you in a very public way, but called their daughter Migsy? Not Margaret but the highly personal Migsy, and they even collared it for their own use as a (probably very commercial) website name for its connotations of closeness to a unique Royal person. Wouldn't you feel that nothing is private and personal to you, that everything is up for grabs?