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This is not a Google rich snippet.
Google calls them sitelinks, and (contrary to rich snippets) authors can’t do anything to get these. So you can’t possibly find anything in the site’s markup.
If it were a rich snippet, you wouldn’t find the relevant data in a metadata file (Google doesn’t support it). Instead, you would find markup on the relevant page(s). You would have to look for structured data in certain syntaxes (JSON-LD, Microdata, RDFa), making use of a certain vocabulary (Schema.org). So you would have to curl
the HTML documents, and use a JSON-LD, a Microdata, and a RDFa parser to extract any structured data.