Hi there,
Thank you for providing that screenshot of what you are seeing.
On my default installation of the latest version of WooCommerce, with the Storefront theme, I do not have a search tab available in the Product Categories metabox:

Screenshot direct link: https://d.pr/i/4OYV7B
To confirm, this is in the sidebar when you are editing a product and applying/adding product categories, correct?
It seems there may be something adding that to your site — either code in your theme, or another active plugin on the site.
To narrow down where that is coming from, please temporarily switch your theme back to Storefront and disable all plugins except for WooCommerce, then check to see that still appears. If it does, then you can switch your theme back and reactivate plugins one-by-one until you find the one that is adding that.
If you are still having trouble, please share a copy of your site’s System Status. You can find it via WooCommerce > Status. Select “Get system report” and then “Copy for support”. Once you’ve done that, you can paste it here in your reply. Please note that since this is a public forum, you may want to remove any sensitive information that you don’t want shared publicly.
I hope that helps!