Microsoft "teamed up" with Skype many years ago to improve what their Messenger app provided. One of the things they did was to 'encourage' signing into Skype with a Microsoft account info. Many sites offer similar sign-in options with your Google and/or Faceblast info, even if they don't 'own' each other.
Skype has owned by M$, since May 2013. So, you've been using a M$ 'product' for the last five years.
Obviously, they would want you to use their "M$" info to log in.
As for as alternates, I would suggest an interweb search (that's how I found the info above).
I find the phone, even a land line works well for me. YMMV.