the WP-CLI always exports to a standardized text file. I have never had issues importing those files. I do use the same mechanism to import the database into the new server. That is ee shell back into your site and then wp db import dbname.sql to get it loaded. You have to make sure you move the file to the htdocs director for the site first and also make sure that owner/perms are such that the shell user, normally www-data, can access the file. Remember to delete that file out of your public folder when done.
As to why you see the icon on your computer, that most likely has to do with your individual computer settings defining which app opens the extension.
EDIT: I am adding the following for clarity.
sudo ee shell mysite.com
wp db export mysitedb.sql
Then move or copy the sql file from the htdocs folder of the origin site, normally /opt/easyengine/sites/mysite.com/app/htdocs to your new server. You can use Filezilla, rsync, rcp or other methods to move the file.
Move the file into the htdocs folder for the site. Normally in /opt/easyengine/sites/mysite.com/app/htdocs folder.
chown www-data:www-data mysitedb.sql
wp db import mysitedb.sql
if you wish to use the phpMyAdmin that is part of the admin tools, follow this guide