Hi,
Thank you for the details! Can you please link a page where the plugin search bar is visible? I can only see the theme search in the header on the front page. Thank you!
If you wish to search the product by product categories, then check this documentation as well.
Also, the WooCoomerce results page is usually not capable of displaying product categories as results – only products or variations. If you are seeing only products there, then it is because of that reason.