if we use the Windows example, it'd go like:
Damn my windows just crashed.
The OS or the browser?
Chrome is a much better name than Doorway or Vents or Tunnel. It's different than Windows. It's fast and shiny.
That aside I guess they're building a brand and want to populate it with some related products. I'm dubious about the value of that approach though; Sun's plastering of the "Java" brand onto on OS didn't do them much good. IBM's WebSphere overloading has caused me problems in the past. Microsoft turned ".NET" from a useful label into a meaningless one.
They called the browser Chrome because part of the mission was to strip back the browser's chrome (i.e. its UI) and give prominence to the web content. This philosophy is likely to be applied to the OS too.
"The OS or the browser?"
The OS only exists to provide the browser, which in turn only exists to provide the web. As others have pointed out, no-one (non-geek) knows what a browser is, I doubt they'd do much better with questions about operating systems.
edit: http://www.youtube.com/watch?v=o4MwTvtyrUQ (What is a Browser?)