Hello Arturas,
Actually that updater only work from the version 3.9xxx update to 3.9xxx or 3.9xxx update to 4.0xxx or above, if you update from 3.8xxx to 3.9xxx, It is not possible to do this via the Updater.. You have to go download the latest version of Easyblog from here
http://stackideas.com/dashboard , then upload/install from your Joomla backend > Extension Manager
Hope this help.