I have a Samsung Galaxy Note 3 running kitkat 4.4.2
I've been running Skype version 188.8.131.520 for years without any issues and recently unable to login anymore, it allows me to enter credentials and seems like its attempting to log in with spinning circle but keeps spinning and never logs in.
The only way i can currently log in is on a browser, either Firefox or Chrome by browsing to the following skype login page
But before entering your credentials, ensure that the browser is set to "desktop" mode in the settings
If you attempt to log in without enabling "desktop" mode you will be diverted to the Play store to install the Skype app
On Firefox you should be able to log in without any warnings about connection security but for some reason on the Chrome browser i was prompted with messages stating that my connection wasn't secure when i know it actually is so i have proceeded and successfully logged in.
I guess Google prefer users to use the app instead of a browser log in method, which skype officially offers on multiple devices that is able to run a browser so it's an alternative to installing an app.