I'm not sure if meta-data tags / name-value pairs is what I'm referring to or not, but I'd like to be able to search Wikipedia's sidebar tags such as the ones displayed in the screenshot below:
In Wikipedia, the information from infoboxes is often retrieved from WikiData. It can be queried there.
If you have your own installation of MediaWiki, you can install Semantic MediaWiki and enrich your infobox templates with semantic annotations. After that, you can make semantic queries against the pages with those templates.