Managing personal Outlook.com account settings, security, and privacy
So this is now solved, and not thanks to the support here, but thanks to the online chat support.
https://support.microsoft.com/home/contact?linkquery=Troubleshooting%20verification%20code%20issues Select start -> from the drop down menu select "Other Product" and eventually you will be queued for getting to the support.
I have to admit, the people there are actually trying to help you (rather than a ton of useless support responses in this forum), and I didn't have to wait longer than 10 minutes (usually it takes 2-3 minutes to reach a person). It took 3 different people to make this work again. So, not sure which of all the steps helped.
The first person asked me to fill in this form, and then tech support contacted me via email that the case was closed, and it should work. Well, it didn't.
The second one asked me tofill out a different form, with much more details, where you fill in some old passwords you used, email addresses to whom you have sent emails, and many more. All of this was to verify that I own the email. But even after the confirmation, we got the very same old issue (Try another verification method). The person on the chat support suggested waiting for some 24h and try again. Some time later, we tried and we got the very same result.
Contacted the chat support again. The third person suggested that it may be a cache issue, so he asked to open an incognito window and try to log in through login.live.com/?nocache=1, and this time I got the verification code, and finally regained access to my email.