Well, Macross Compendium do use the surname "Linn" for everyone of her family except herself. Weird. The old page did use both version of that romanization.
On a side note, if her name is Chinese, then Ling would be the closest romanization for her surname. And Chinese usually use "i" rather than "y", anyways, so I'm not too surprise that Linn is also used as the alternative romanization.
Because there's no such surname as 鈴 in Chinese, although 泠, similiar character with the same sound, is indeed a surname, although it's rare one. 林 is a very common surname in comparison, hence the confusion.