Thank you all for your responses. A problem shared is a problem halved.
I thought I screwed up my phone, but if Samsung just screwed up and this will be fixed soon, I'm actually happy.
I managed at last to sign in to my Samsung account!
I don't know if it will work for any one else, but I did as described in the text that follows.
I choosed to use the watch without connecting it to the phone.
Connected the watch to my WiFi network and then I signed in to my Samsung account on the watch.
Then I started the galaxy Wearable app on the phone and initiated the process of connecting the watch to the phone.
When the blank white log in screen appeared I backed out from it, and suddenly I got the message that I was logged in on my Samsung account...
After that everything has worked as it's supposed to!
I hope that this can help someone else with their problem of connecting the watch to the phone. 🙂