Skype web client is not possible to create by anyone except Microsoft.
Skype offers scriptable API, but this is inside the desktop client. The Skype protocol is highly guarded secret.
The only way to accomplish native "web skype" would be to run one Skype application + full desktop environment for each web users (basically start a virtual machine per use). This is too heavy for any service to scale.
My guess is that web clients use older MSN chat protocol to discuss with Skype (after Microsoft acquisition Skype started to support it). http://news.en.softonic.com/skype-integrated-with-msn-messenger