This is why I use SoundHound. It'll pull up lyrics and give you the option to open in Spotify or Apple Music. Not sure if Shazam does this, but when I open the SH app, it starts listening. I don't have to wait for the app to open, and then click listen.
Cortana music identification only works for music playing loud enough for your microphone to hear it.
There are other apps that let you identify music:
Shazam for Windows 8/10 Desktop: https://www.microsoft.com/en-us/store/apps/shazam/9wzdncrfj0qq
Shazam (iOS, Android and Windows Phone) http://www.shazam.com/apps
Soundhound (iOS, Android, Windows Phone and Blackberry): http://www.soundhound.com/soundhound