Annie is talking to his fake Dad & telling him about his real Dad character during wedding anniversary before Annie's mother
Annie: Dad always does something really special for the anniversary.
Fake Dad: Like what?
Annie: One year he had a star named after her.
Fake Dad: He had a star named after her? Well, that's nice, but isn't that a little... corny?
What do you mean star named after her means?
Is Annie saying her real dad calls her wife with star names during their Anniversary?
for example: Hi, Miss Beauty Proxima Centauri.