If the person says she listens to the Japanese anime songs, she might listen to JPop later! Maybe that person doesn't know much about Japanese music and thinks anime songs are typical jpop songs!
I started to listen to Japanese music because of animes! I really liked some openings so I searched for more songs of some bands and soon I was really into jpop and tried out many other artists (not only "anime artists").
My point is, if that person is interested in anime songs she might try out other jpop songs and becomes a fan. But I also think to say "I'm a jpop fan" at that point is kind of wrong, the person should have been more specific, like "I like japanese anime songs".
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