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Please open Joomla's configuration.php file, located in the site's root, with a text editor. Find the $dbtype line. Look at the right hand of the equals sign, inside the single quotes. If that contains mysql please change it to pdomysql

Why you have to do that. Old versions of Joomla! would by default install using the 'mysql' database driver. This is an old database driver deprecated with PHP 5.3 and made obsolete (removed) in PHP 7.0. The recommended driver changed to 'mysqli' which is newer and better. There is a third driver 'pdomysql' which uses the PHP Data Objects (PDO) extension for PHP. This is considered an even better, if more cumbersome, alternative.

Your site very likely had the old 'mysql' driver. This means that upon upgrade to PHP 7 your site would immediately break. Joomla! tries to prevent that by understanding that using the nonexistent 'mysql' driver under PHP 7 really means using the 'pdomysql' driver. Therefore it internally uses the 'pdomysql' driver BUT reports to its extensions that it's using the 'mysql' driver.

Akeeba Backup sees that Joomla is using the 'mysql' driver on PHP 7 which is, of course, impossible. However, Akeeba Backup's default fallback in this case is 'mysqli', not 'pdomysql'. This has to do with compatibility with some third party extensions, such as Falang, which do weird things with the database driver. The problem is that your host has not enabled the 'mysqli' driver on PHP 7.1, thus causing Akeeba Backup to fail.

By changing your configuration.php file to use 'pdomysql' you are essentially giving Akeeba Backup the hint to use the correct database driver itself, therefore fixing the problem you have.