Very short description of key items to check:
1. Hostname is correct. It is better to use IP address rather than domain name.
2. Your hosting allows direct MySQL connections. As for example GoDaddy does not.
3. You have specified correct MySQL port. In most cases it should be 3306.
4. Username and password are correct.
5. User has enough permissions for used database. It should be configured in website management panel or in phpMyAdmin.
6. Database name is correct.
7. Your IP address is allowed to connect to MySQL (included in Access List) – should be configured in website management panel.
8. You have no firewall or similar software blocking Store Manager connection requests.
Important note: If you are using Store Manager for PrestaShop and unable to connect to more than one store, most likely that you have Standard Edition of Store Manager.
In PrestaShop Store Manager the number of Store connections depends upon the edition. Please, check the link for more details: www.prestashopmanager.com/order/. Thus to manage more than one store, you need to upgrade to PRO Edition.
When all of these items are checked you should be able to connect. There are no specific limitations in DEMO nor other specific actions required to establish connection.