Save search criteria per user and show on custom user page

Assuming that those users have accounts on your site (are subscribers, authors, or some other role), you can add custom fields to their profiles. The easiest way is to use a plugin, like Advanced Custom Fields (https://wordpress.org/plugins/advanced-custom-fields/).

The fields would be available in their profile pages, and you can add, and use in any template.

I would give you code examples, but their site seems to be down… I’ll try to update the answer once it’s up again.

Update: ACF could take care of creating the fields, and displaying them in the user profile page, and you can use add_user_meta to save them:

add_user_meta( $user_id, 'custom_field_name', $search);

If you don’t want to use ACF, you can display the fields in the profile with custom code. Deleting the searches would be a bit harder, though, but doable.