Which is something that's in contention by itself. Rin's a technophobe. While she does watch TV occasionally I think, it's not at her own place and she doesn't have a TV of her own if I remember right. Plus she doesn't strike me as someone who's too concerned with pop culture stuff, especially something like Dr. Who that's main fanbase is in England.
It's not that having a character making references is bad and that it should be never done ever, it's that it has to make sense for that person's character, and when that sort of character starts making a bunch of references out of nowhere outside of comedy or parody, it's both OOC and a bit irritating. For Rin, it doesn't make sense for her character to just be dropping references like that, and that's the main issue.