How to log out from skype app?

The Skype account has been linked with your Microsoft Account. At the time, there is no way to remove this link.

Found this on Skype forum: Unlinking Microsoft Account

Please contact Skype customer service as they can help you unmerging Microsoft accounts that were merged with a Skype account incorrectly.

The page Why is Skype for Windows 8 always on? explains that you can't, because this way, Skype can be used as a phone.

The best way to answer those questions is with a metaphor. Imagine you had a mobile phone but every time someone wanted to call or text message you they had to find a way to tell you to turn on your phone. It wouldn’t be very useful, if they had a way to tell you to turn your phone on, they wouldn’t need to call you.

While using Skype press Win+C (charms bar) then click Settings and in app Options you will be able to logout.