Hi,
The filters system is based on using a products listing to display the results. And a product listing can only display products, not variants.
So unless there is a complex and profound modification on the way the filter system works, it won't be able to search through variants.
Regarding the variant search with the Joomla search, it depends on many parameters.
For example, if you've set a canonical URL in the product and have the force canonical URLs on listings, then the canonical URL will override the dynamically generated URL and in that case, it will display the default variant. But if you don't force the canonical URLs, then the product page will display the default variant being the variant of the result.
Regarding the search, I don't see why it would work on other fields of the variants. There must be something else I'm missing from your explanations. And for the name being displayed, what name did you enter in the variant ?
It would be great if you could provide a screenshot of the settings of the main product and of the variant setting for your search example so that we could better understand the situation.