The apps do, but the website does not. music.google.com used to have native support, but to cast music.youtube.com you have to cast the tab, which is far less than perfect. (Instead of streaming directly from the CDN to the chromecast, casting a tab will stream from your desktop to the chromecast. It works so poorly that sometimes even the pitch changes slightly, like an old timey recording.)

It does have Chromecast support, I've used it at least on Android and on music.youtube.com. I can't vouch for iOS though!