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!
over and out!
And finally I get this :
Your browser is not supported. Please update your browser to newest version.
I can't update because opera 36 is last version for windows XP
I can't update my OS either because updating means I will lose all my bookmarks
So I'm stuck I hope you all happy with this because certainly I'm not.
When you install Opera, in the installer, you can click "options". There you will see an option to import data from your default browser. It's actually checked by default. If you choose/keep that option with Chrome as the default browser, Opera will import Chrome's passwords etc. and not just bookmarks.
If Opera is already installed, you should be able to goto the URL opera://settings/importData in Opera, make sure passwords are checked and import from Chrome.
Another way to do just passwords is to export them from Chrome via chrome://settings/passwords, turn on opera://flags/#PasswordImport in Opera and import the password file by going to opera://settings/passwords, clicking the kebab menu on the side (3 dots) and choosing "import".