Why does the YubiKey Neo only use Firefox, not Chrome? I want to use the YubiKey NEO with Chrome, and when I tap the NFC sensor to the phone, the only app I can use is Firefox.

I would like to have SSO work with Chrome, but not sure how they were able to make Firefox listen for NFC events.
Have you tried clearing your defaults for your browser? If you have Firefox set as your default browser, anytime anything on your phone makes a request that needs to be handled by a browser, Firefox will start up by default with no other options.

You can easily tell if Firefox is set by default by tapping a link in another application, such as email or Facebook.  If you are presented with a list of apps to choose from to open the link (e.g. Firefox or Chrome) and 2 buttons - one to use the chosen app every time to open links moving forward, and one to open the link with that app just this one time only - you have no default browser set.  If instead when you click on the link Firefox opens immediately, Firefox is set as your default.  This could be what is happening here with your YubiKey.