NOt Available of Search box and menu, mini cart at Checkout page is Magento 2 default feature and depend on page layout. There are not related to config setting or multistore-setup.
Magento2 create a specific layout for checkout page layout="checkout"
.
See at checkout_index_index.xml
<page xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" layout="checkout" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_configuration.xsd">
And that layout checkout has been defined at Magento_Checkout
<layout xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_layout.xsd">
<update handle="empty"/>
<referenceContainer name="page.wrapper">
<container name="checkout.header.container" as="checkout_header_container" label="Checkout Page Header Container" htmlTag="header" htmlClass="page-header" before="main.content">
<container name="checkout.header.wrapper" label="Checkout Page Header" as="checkout_header_wrapper" htmlTag="div" htmlClass="header content"/>
</container>
</referenceContainer>
<move element="logo" destination="checkout.header.wrapper"/>
If any demo store using the checkout
layout thenn that demo site checkout does not have any Search box and menu, mini cart .