On the site, click the badge at the left of the address field, goto "site settings" and allow popups and redirects.
Then, at the bottom of the page, set the max to 50. Then, click "choose file" and point it to your session_xxxx file. Then on the bottom bar on the page, click recently closed (don't worry about the name "recently closed") and then click the top entry to open all the URLs in tabs.
If you need a max higher then 50, right click the max field and choose "inspect element". Then, on the Elements tab in the developer tools, you can right-click on max="50" on the max tabs input element, choose "edit attribute" and change the number to something higher. Then, hit ctrl + enter to apply. Then, you can increment the max to the new high, choose your sessions file again and then open all the tabs.
Well... As Mac-2 is on the way to repair service, I decided to purge the HD and reinstall the MacOS operating system, altogether. With Mac, its not such a big deal as with Windows. But one only needs to do it in very rare occasions - like now, when I'm testing a heating problem of Mac-2.
After the OS was installed, I downloaded Opera and logged into sync. And now I can see all three devices - Mac-1, Mac-2 and Windows-PC open tabs, bookmarks and speed dial.
I guess we can call this case solved, even though we don't know what caused the problem, initially. Thank you @lecog for your help, and let's carry on from here. Have a great summer and stay healthy!