In the past we supported three families of database servers: MySQL (MySQL, Percona, MariaDB), PostgreSQL and MS SQL Server (MS SQL Server and Azure SQL Server). However, our usage statistics showed that the number of users for non-MySQL servers were very literally less than a handful.
At the same time non-MySQL databases do not have good support for exporting the DDL of their tables and views, much less for procedures, functions and triggers. We had to go to extreme lengths to provide a mostly accurate database dump. Because of that they also required a lot of very complicated maintenance which cost a disproportionate amount of development resources compared to their user base.
For this reason we decided to remove support for database servers not in the MySQL family in July 2019 with the release of Akeeba Solo and Backup for WordPress 3.6.0 and Akeeba Backup for Joomla 6.6.0. There are no plans to reintroduce support for them.
As you can see in Akeeba Solo there is no longer an option to set the Database Driver to anything but MySQL, MySQLi, PDO MySQL, None and SQLite (the latter two are basically the same since SQLite is in fact a file that's already being backed up). While nothing stops you from entering the connection information to a PostgreSQL database server there it will most definitely not work since PostgreSQL and MySQL are completely different database servers which use different connection protocols and different SQL dialects.
Therefore the message you are receiving is correct. Akeeba Solo only supports MySQL-family database servers and you're trying to use a PostgreSQL server. They can't talk to each other.
Nicholas K. Dionysopoulos
Lead Developer and Director