Add product as a filter into the search

Problem

Searching in the docs is a bit confusing because:

1. it's not clear which product the found result is for and needs to be opened first to figure that out
2. it's not possible to filter the results only per specific product

If I look for some docs for oCIS, oC 11 docs are popping up and I don't know it until I open the result. Then I need to search again and if there are multiple results, it can take a few attempts until I reach the correct place.

Proposed solution

If possible, it would be great to:

1. Show a small badge per search result row with the product (oCIS, oC11, desktop, etc.) to immediately see what the result is for
2. Add a small dropdown to the search where a product can be picked and used as a filter (default to: oC main page -> all, specific product -> that product)

Start by locating the existing documentation search entry point and how product information is represented in search results. Determine where result rows and search controls are rendered, then verify the desired default and product-specific filtering behavior against the issue description. Done means results show their product and users can filter by product.