We don't hear often how Google classifies results Featured News nor how related news is chosen for desktop and mobile users. Usually when someone from the search team comments on news results, they say that they use their own algorithm for news.
However John Mueller at google Webmaster Office Hours offered some insights into how Google determines which stories to show and how they rank them.
The “News” and “Top Stories” sections of search results are completely algorithmic and based partly on content, partly on queries, partly on how we think it makes sense to display websites with those types of stories there above.
Part of this I think also depends on whether you're using AMP on your pages, valid AMP or not, so all of these things kind of come together. Essentially, it is organic ranking and this determines which URLs we show and in which parts of the search results.
Specifically in regards to these result boxes and different parts of the search results it really depends on a number of different factors. So there is not a single thing that always appears there or never appears there.
The “how it makes sense” is the part that most SEOs are thinking about. Since most news does not have a direct link to that news, it would be a bad sign, although the value of the link to the site could be used. AND appearing on Google News is not an exact science either, since we have seen stories in these sections that are not in Google News, or at least are not in good positions.
Personalization could play a big role here, more than it could in Google News.
John also talked a little about the role of AMP in this section.
And some of these depend on markup perhaps, or if you're using AMP, some of these only make use of AMP results, so that's something to check, if you're trying to display AMP and something is technically wrong in the way you're serving AMP pages, then that's something you need to fix as well.
It's the AMP results that stand out in the top stories section in mobile search results. So if you're trying to get into this section, AMP implementation could make a difference.