I believe the connection status as reported by that icon reflects what the Windows "Network Awareness" (or NCSI) facility thinks that the state of the system's Internet connection is.
In my experience, it doesn't always reflect the true state of the system's Internet connection.
I've found if you disable it (using the registry edit found here) it produces similar behavior on a laptop I have - it does not say "Identifying..." but it does correctly identify my wireless SSID - and yet says "No Internet Access." I also experienced weirdness with that icon when experimenting with my own Linux-based DNS.
I'm not aware of any programs that perform the equivalent of that registry edit. Furthermore, according to that linked Microsoft KB, it says programs that rely on NCSI might not work properly if NCSI is disabled. I don't know of any that do rely on that. I've not experienced any adverse effect on the laptop other than the erroneous message.