Note: This is yet another of my short throw-aways.
The problem with this response is that the people who give it almost certainly never listen to Hungarian folk songs or Baroque orchestral music. They almost certainly never listen to bebop or children's songs. It is highly unlikely that they listen to Inuit throat singing. A good exercise is to put it to them as a follow up question: "Oh, really? So you like bebop? What's your favorite from Coltrane?" See how far you get with them.
From this, we can probably conclude that the phrase "anything but rap and country" is shorthand for "anything that does not sound particularly idiosyncratic." It's not that anyone must enjoy rap or country, it's just that if one doesn't enjoy any rap at all, and doesn't enjoy any country at all, then it is probably not true that they enjoy "anything" else. More probably, what they enjoy is popular radio music: classic rock, "80s, 90s, 00s, and today," and so on. What they really like is unremarkable music that does not require that they pay a lot of attention to what's going on.
Even musical styles like Mexican rock music or Bollywood pop music are unlikely to impress these folks who profess to like "anything," and Mexican and Indian music are both highly similar to what you find on North American radio, for the most part. A good Indian club song is not all that different from a good American club song. A good Mexican rock song sounds like it could be played by Hinder or Slash or 30 Seconds to Mars, or whoever happens to be the it-band of modern rock these days. But the fact that the singers sing in languages other than English means that a non-English-speaker is marginally more distracted than he or she would otherwise be.
In short, anything that demands more of a person's attention than the run-of-the-mill stuff is what these people actually dislike. If it sounds somehow different - foreign language lyrics, instruments uncommon in modern pop/rock, scales or chord changes that do not fit in with the Rolling Stone paradigm, etc. - it is likely to distract them, and being distracted means they don't like it.
Rap music is obviously idiosyncratic. There might be a little singing, but it doesn't tend to sound like Katy Perry. The beats can be somewhat jarring and aggressive. The poetry is often bellowed instead of crooned or even softly spoken. And so it goes with country. Although country music is less aggressive, it is no less idiosyncratic, with its fiddles and pedal steel guitars and the unique accent/diction of its singers. It is just different enough to be distracting.
When someone tells me that they enjoy "anything but rap and country," I make a mental note that the person is not much of a music fan. Can you imagine someone saying that their favorite type of painting is "anything except cubism and late impressionism?" Can you imagine someone saying that their favorite type of book is "anything except murder mysteries and historical fiction?" Can you imagine someone saying that their favorite kind of dancing is "anything except clogging and polka?"
Indeed, all you really know about people who say such things is that they do not have a large investment in whatever it is they're talking about. And they don't really need to be heavily invested in music, either, but then why do they make the claim that they like "anything?" They don't.