The theme ajax search function only searches in titles and descriptions. To search by other attributes like SKU you have to use a plugin like https://wordpress.org/plugins/advanced-woo-search/
To use any plugin that replaced the default search box, first you have to disable the ajax search feature in Appearance > Customize > Header > Header Icons
Then, in the Advanced Woo Search settings enable Seamless Integration
That’s the basic configuration.
Then, you can check the search sources