Skype simply needs to update its app. Users have posted these workarounds while we wait for them to do that:
Read this post: 10.8.5 Broke Camera Usage For 3rd Party Apps Like Skype
It's not so much that Skype should update its app, although it would indeed be better that it is 64-bits.
The main issue is that the 10.8.5 update broke compatibility for 32-bits apps, when older and even newer versions of OSX still have are still compatible.
I'm not 100% sure of the point you're trying to make, but if you try these apps, you'll see that the camera doesn't work either :-)
They are Apple apps, but they are 32-bits and thus have no access to the Facetime HD Camera on Mid-2013 MacBook Airs
The main issue is that the 10.8.5 update broke compatibility for 32-bits apps,
Whatever the problem is, it's not the fact they are 32 bit apps, since others continue to work (add Firefox to the list).
What else isn't working for you?
Have you tried the workarounds in the threads I posted?
Sorry I was not clear. What I meant is that in the 10.8.5 update, Apple broke the ability for 32-bits apps to access the Facetime HD Camera on Mid-2013 MacBook Airs.
This is because the AppleCameraPlugin that is installed with 10.8.5 only has the 64-bits version, whereas 10.8.4 and newer (beta) OS have a universal version (32-bits and 64 bits) of that plugin
Skype is still able to run, it just doesn't have access to the camera. Just like iPhoto, iMovie, Google Hangouts, etc.