Manual:$wgCookiePath

Details
If you do not want your login to be persistent across Wikis running on the same server, and your installations all use the same database, you will need to modify the LocalSettings.php file $wgCookiePath = " /directory "; Where  is the directory for that MediaWiki installation relative to the DOCUMENT_ROOT.

This resolves issues where a user is occasionally prompted to log in when viewing a different installation, even though they logged in earlier and asked to have their password remembered.

$wgSessionName and $wgCookiePrefix provide alternate solutions.