Go down to "View my account", and press A. Highlight your profile on the left of your screen, and press "A" (It will either have your profile picture or the blank grey head with a green status light over it). This walkthrough assumes you are already logged in under account A, and want to log into account B instead. Unlinking the accounts logs you out, signing in to a new account links the new account, and thus signs you in every time you start the app.
The Xbox One logs into Skype by looking at your Microsoft account and seeing which Skype account is linked to it. My best guess is MS and Skype support are assuming you have already tried this process and it didn't work, and that's why they are not giving you this (easy) answer. Anyway, I decided to make a stand-alone post so that those searching would have a step-by-step guide when (if) they searched for one. Honestly, I thought the process was very simple (albeit a little long-winded) although it could be streamlined quite a bit. These issues always seem to come with a "I contacted MS and Skype support and they both said it is impossible" or something to that effect. I've seen quite a few posts regarding Xbox One's Skype app and issues with signing in and out of different accounts on your gamertag/Xbox One.