The problem is that I must press "log in"/"sign in" button on most web-services again and again per day. All login/passwords are stored successfuly but the feature which is often called "remember me" doesn't works. What could be the reason for this bug? In before: the cookies are enabled.
UPD> I must say that I have an iPhone and an iPad with enabled keychain sharing via iCloud but the problem I have on the mac never happens on my devices. Maybe it will helps.