Is cookie information stored on the server side?

I thought cookies are stored on client side only, as files. Then I realize if cookies are not stored on server side, how could a server match a cookie just received from a client to some session or other information created in the past?

So is it correct that cookie information is also stored on the server side?

How does a server store cookie information?

I have searched about “client side cookie” and “server side cookie” and reached several discussions including