Hi @Johannes95
Your home page is only providing structured data for Product that looks to be coming from featured products, you should provide Organization and WebSite if you are wanting to alter your search results, image and description etc.
You can check with the following tools
https://search.google.com/test/rich-results
http://linter.structured-data.org
Schema Definitions
https://schema.org/Organization
https://schema.org/WebSite
https://schema.org/Product
There should be something like this in your header.liquid ( this is the default from dawn theme )
{%- if request.page_type == 'index' -%}
{% assign potential_action_target = request.origin | append: routes.search_url | append: "?q={search_term_string}" %}
{%- endif -%}
Regards