OK, when you said 5.6 I misunderstood. MySQL 5.6 should work just fine, at least to the degree that any software installed in the site you backed up doesn't use any MySQL 5.7 or 8.0 features. It would definitely have no impact over the connection to the database.
The database connection code using in ANGIE is actually the same code used in Joomla itself. So it's not an Akeeba Backup issue or your site itself would not have been able to connect which means that you wouldn't have been able to back it up. Your test was correct.
If you search the MySQL error code (2002) you will see that it means that MySQL cannot connect through a socket file. Instead of using localhost use 127.0.0.1 for the database server hostname. There is a massive difference between the two in the context of connecting to a MySQL server as you can read in the page I linked you. localhost means that you want MySQL to connect with a socket file, 127.0.0.1 means that you want to connect through TCP/IP.
If this doesn't help it would mean that you have disabled TCP/IP networking in your my.cnf file and that you have either disabled socket connections or your socket file does not have adequate access permissions for other processes to connect to it. A final possibility is that you installed the MySQL client package but not the MySQL server package — I've made that mistake myself but at least I knew how to correctly check with aptitude
what the heck I've installed :p
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!