Lily is a name in it's own right (from the flower). I'd be surprised to meet a Lily who was really an Elizabeth, and think you can have both without that being an issue. Elizabeth has so many diminutives, I really don't think it is at all like Kate Katherine (to me the equivalent would be more Lizzie Elizabeth).
It is very popular though, and yes perhaps too many 'l's. Depends a bit on your surname whether that's an issue, I don't think people often use the full name without the surname