It seems like your database account (or wordpress) does not have sufficient privileges to create database tables. The reason why you don't see the search instance, it's because the neccessary tables are missing, which should have been created when you activated the plugin.
You have few options here:
1. Create the tables manually. I suggest exporting the database tables from your local MAMP installation and importing them on your server via PHPMyAdmin (or your prefered tool). The plugin uses the following tables: wp_ajaxsearchpro, wp_ajaxsearchpro_priorities, wp_ajaxsearchpro_statistics, wp_asp_index
2. Checking and fixing the database user permissions in your CPANEL. I highly recommend doing this, as many plugins need permissions to create database tables. Then reinstalling the plugin should solve the problem.Best,
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