@Randomizer No clue, probably as a safe-guard for when nosy housewives (I really hope those are just a jp trope and not a jp reality) ask him why she lives with him, or why she has elf-ears.
No, really, I don't get why he can't just make the story that she is an overseas friend either.
Also, when he had to explain archaic jp lifestyle to her, I wondered why he couldn't have put more effort in finding something that could implicitly teach the actual level of technology better. Though I suppose it is really hard to find an educational show that is present-day (as opposed to centuries past) without having a mad scientist and plot-devices popping up that use completely made-up science. So probably safer to go for the archaic ones.
In the end, did she just want to watch the song? He should have asked, and if so, put youtube on a loop for the OST.