So I figured out that the breeds are stored as product tags. And that's the solution, you need to search product tags instead of products.
I've turned off the search in posts and pages and products and on the General Options -> Sources 2 panel I dragged the product tags as the source: https://i.imgur.com/ftIg5KY.png
Now I'm getting results like this: https://i.imgur.com/gPyX92M.png
I hope that's what you were looking for.Best,
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