Does any browser in iOS support push notifications similar to Chrome on Android? Google seems to mention it in their GCM presentation video but I can't find any information about it.
There currently aren't any browsers on iOS that support the Web Push Notifications feature Chrome for desktop and Android do— or Service Workers, as you rightfully point out.
See the implementation status of Service Workers in different browsers.
Apple has experimented with push notifications on Mac OS X, documentation for which is available here.