No, you cannot share cookies across web browsers. At present, there
are no services that synchronizes cookies just like how bookmarks are
It does not make sense for you to share a cookie across
identify unique sessions. Most web applications are also intelligent
enough to handle simultaneous access of their service from multiple
devices. For example, you can use Facebook or Gmail on your smartphone
and laptop without causing any synchronizing issues.
There are some google chrome extensions that might intrest you. For example take a look at this extension:
Create an account. Log in into multiple devices and quickly share you browser session. It's working pretty good so far. The sync time is almost instant and it is very nice and easy to use. Yes it does do the sessions per website but adding a new website session is not hard at all and doesn't take alot of time. They also have a paid version which is only $1 per month and it gives you some extra things.
On thing I am missing with this extension is that it doesn't share cookies / passwords so I don't have to login again on every site. I searched some more but I couldn't find anything that shares cookies/passwords automatically. I hope sessionbox will implement that feature sometime.