How to set Skype Name in newest Skype
I ran into the exact same problem as you today and found it highly frustrating there was no way to choose a username for a Skype account like before. Even thorough Google searches on the topic turned up nothing of use on the topic.
I managed to find a solution, even though no instances of Skype on any of my machines available to me would present me with a sign up screen which allowed me to choose a username so I had an idea to try sign up via the Skype app on my mobile device and miraculously it presented me with an option to choose a custom user name! Therefore this is (at the time of writing) a working way to sign up for Skype with a username of your choice.
Skype for Windows 7, 8 and 10 and Skype for older versions of Windows system are two different programs. First was created / build, when Microsoft has already purchased Skype, while Skype for Vista, Windows XP and older systems was build before selling Skype to Microsoft.
For this reason, if you click Create new account
in Skype installed on for example Windows XP, you will be redirected to "old" Join Skype page, where you can pick your own Skype Name.
If you're using Skype for Windows 7 or newer versions of Windows, you are forced to create account on Microsoft, not on Skype and use that account in Skype and thus you can't pick your own Skype Name. You must use either auto-generated one or your e-mail address, used during registration.
The 'Create an account' sign up process is directed towards signing up for an account using either a phone number or a Microsoft account as your alias.
This link will allow you to pick a Skype Name on sign up instead: https://login.skype.com/registration