The BBC has a unique exception when it comes to music, I believe. That’s why you often hear The Beatles in the Eastenders cafe. ITV couldn’t do that.
Yes they can, all they need to do is stump up for the licenses the same as the BBC do via organisations such as the PRS:
https://www.prsformusic.com/licences/broadcasting-music-on-t...
https://www.bbc.com/academy-guides/music-copyright-in-tv-wha...