So… My assumption was right: I got the chrome extension (of LastPass) on Windows and the opera extension on Linux.
On Windows, I removed the chrome extension and installed the opera extension.
And now… I notice the synchronization icon that I never saw before. So I think that, indeed, there were a synchronization issue!
I'll see in the next if I still have some troubles when switching from one instance to the other!
Thank you for your help @leocg !