Hello,
I have test your site and noticed that you are using two search elements one for products and second for posts as in the below screenshot: https://jmp.sh/4TNFDPU
As the mobile search is shown via the mobile menu element and don’t have the option to select the post type so it inherits the “Post Type” of the desktop search. As your second search have the “Post Type” as “Post” and not the “Product” that’s why your mobile menu inherits it’s functionality and display “search for posts”.
Possible ways to display the mobile search for products and not for posts:
1) You can change the post type of your second search element to “Product” in desktop layout.
2) You can delete the second search so the mobile menu search inherits the first one.
3) You can change the sequence of searches so if you used the “search for products” as a second number the mobile menu inherts this one because it is second in sequence.
Result: https://jmp.sh/Ro72cH9
Best Regards.