She actually said many times she listens to classical music. By the way, many artists don't listen to much music around because it sometimes causes them to "steal" ideas even subconsciously. By the way, not listening to music at all creates a lack of culture in a singer.
I think her statement is not to be taken too seriously
Of course you can be an author without reading a lot or a film maker without watching lots of movies. Same with a musician who doesn't listen to music much. It doesn't mean they didn't for a huge part of their life but if you consume that much when you're younger it can often get a bit repetetive later on, same stories, same sounds. So you're more selective with what you read/watch/listen to.
I think when it's your job as well you probably don't get the same enjoyment from it. She won't want to accidentally copy ideas for a start.
From what I've observed on her insta stories, she's been listening to more stuff now, popular stuff. Billie, 5SOS, Shawn, that Japanese rock band Ok go or something, Maren Morrris, that Bebe Rhexa song, I think Halsey she likes too, maybe Travis band, she likes Nicki. Before I found it odd too how she only said Blink 182, jazz music, and Radiohead. So now I think she's opened up to new music.
I don't think she means she never listens to music, probably more like when she's in the process of making an album she avoids listening to too much stuff, especially stuff on the radio, to avoid copying anyone. We know that she enjoys listening to a lot of jazz, for instance, like when she is just cooking at home and stuff. But I think it's a good thing that she when she is writing her own music she tries to avoid a lot of outside influence because not only does she avoid sounding like anyone else, but it means that she is able to come up with the most organic, original music she possibly can because it's all coming from her own ideas and not what she's hearing others do.
But i think she was so tortured and criticised on her early years being called punk and not knowing who the sex pistols were that she refrains when the topic is "what music do you listen to?" I guess she doesn't want to be in that page again so she just holds back.
I can see that as a reason. People can be so savage about anything. I often see it in music especially rock fans. Like "how can you say you're a fan if you don't know all their songs and history". Those kind of people suck.