@burnout426: I created a completely new browser profile and unfortunately problem persist: all site's cookies are gone after I close the browser.
Here is info from terminal:
$ opera-developer
[63590:63605:0704/172305.290416:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63590:63605:0704/172305.290624:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63590:63605:0704/172305.435458:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63590:63605:0704/172305.898763:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63590:63605:0704/172305.902844:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63537:63537:0704/172307.979459:ERROR:CONSOLE(1950)] "Uncaught (in promise) Error: API not supported", source: chrome://settings/settings.js (1950)
[63590:63605:0704/172325.809093:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63590:63605:0704/172325.809192:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63590:63605:0704/172326.778455:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63537:63537:0704/172333.104203:ERROR:atom_cache.cc(231)] Add chromium/x-internal-source-rfh-token to kAtomsToCache
[63590:63605:0704/172333.484991:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63590:63605:0704/172333.489399:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63590:63605:0704/172333.625477:ERROR:ssl_client_socket_impl.cc(880)] handshake failed; returned -1, SSL error code 1, net_error -202
[63537:63537:0704/172418.848022:ERROR:CONSOLE(503)] "Uncaught (in promise) Error", source: chrome-extension://ebongfbmlegepmkkdjlnlmdcmckedlal/touch_communication.js (503)
[63537:63537:0704/172420.545800:ERROR:CONSOLE(503)] "Uncaught (in promise) Error", source: chrome-extension://ebongfbmlegepmkkdjlnlmdcmckedlal/touch_communication.js (503)
[63537:63537:0704/172420.558850:ERROR:CONSOLE(503)] "Uncaught (in promise) Error", source: chrome-extension://ebongfbmlegepmkkdjlnlmdcmckedlal/touch_communication.js (503)
Here is the same thing as with my last post: last 3 error messages appear at browser close event. I didn't sing in to my opera account so sync is disabled. Never mind handshake errors - I use self-signed SSL certs for local development.