I have no idea then. I searched all of the documentation about translations, but there is no more information about this anywhere. The translations are automatically managed by wordpress – in this case Locotranslate also does some enqueueing too. The translation was generated via wp-cli, so it must be okay. I cross checked WooCommerce and other popular .pot files, and the headers seem to be very similar, only the plugin name differs.
Accoring to the documentation, you can use the system path to place the .mo files outside the plugin folders, and WordPress will automatically recognize them.
I tested that by translating the plugin from the original .pot file, generated the new .po and .mo files, moved to the wp-content\languages\plugin\ajax-search-pro-xx_XX.po/mo location. Then I switched the language, and everything worked just fine, the translations worked.