I had a similar experience, the first time I logged into Skype it asked if I want to save the password in my key-ring which I declined. The next time I started my PC it asked me to login again. What I found with my experience is that if Skype was set to auto-start at login then it would login fine automatically each time I rebooted my machine.
Recently I disabled the auto-start feature and found that I it asked me for a username and password the next time i started my machine. It does not concern me too much if I have to login each time, however I suspect it could be due to something with the key-ring but that is just my assumption.
I agree with the above posts that its best to be left logged out when you are not using the application and I also understand that it is your choice to have it logged in automatically.
I hope my reply can possibly help guide you in a direction in order to find a solution that best serves your needs.
It was performed on a Debian 10 Buster machine
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