Oh boo hoo. He's promoting his own internet radio show, it's up to him to work out clearances. He asks what basis they have for doing it, like choice of law etc., well read the ToS, that's why it's there. 'Other people do it too!' is about the most juvenile argument there is, and the fact that they haven't experienced problems yet is neither here nor there. Maybe Universal hasn't got around to yanking stuff from them yet, maybe they have proper clearances or pay ASCAP feees, or maybe Universal/Soundcloud is enforcing their copyright arbitrarily...which they're perfectly entitled to do as a private entity. Equal treatment under law only applies to the government, unless you can show you're being discriminated against as a member of a protected class.
So his problem is he doesn't know which song(s) is/are infringing? Take your tracklisting and check it against discogs.com or some other music publishing database, and see which songs were published by Universal Music. Maybe 30 minutes of work for a 1 hour show, tops. The responsibility to check clearances does lie with the person who wants to use a copyrighted work.
Edit: I realize this is an unpopular view, but if you're downvoting I invite you to spell out what you think is wrong about it. Every magazine/website/course aimed at budding record producers/DJs/musicians provides coverage of copyright and licensing issues, to help their readership collect what is owed to them as much as to avoid infringement. There is no way you can be an aspiring or actual professional in the music industry for any significant length of time and not be aware of your basic obligations in this area.