Hi Lucas,
Thanks for writing in! Your issue seems to be directly related to server configuration issue probably caused by PHP running as the apache user (mod_php or CGI) rather than the user that owns your WordPress files (suPHP or FastCGI). You can ask your hosting provider to switch over to using suPHP or FastCGI would probably fix it.
As an alternative solution. could you please try adding your FTP credentials directly to your wp-config.php file, to bypass this issue. You can also read more information from here (https://codex.wordpress.org/Editing_wp-config.php#Override_of_default_file_permissions).
define( 'FTP_USER', 'replace_this_text_with_ftp_username' );
define( 'FTP_PASS', 'replace_this_text_with_ftp_password' );
define( 'FTP_HOST', 'ftp.your-site.com:21' );
You can add this before the following lines.
/* That’s all, stop editing! Happy blogging. */
If you’re still having any issues, add the following code fragments also into your wp-config.php file to set the correct file & directory permissions.
define('FS_CHMOD_FILE', 0755);
define('FS_CHMOD_DIR', 0644);
Hope that helps.
Thanks!