Some streaming boxes will do exactly that; both Apple’s and Amazon’s will, and perhaps others. For example, if you search for a movie or TV show on Fire TV or in the Apple TV app (on that box) it’ll show you the title and what services currently have it available. Searching for Stranger Things and House on Apple TV, for example, return links into Netflix or the Prime Video app (where I’ve been watching the latter).
In the US at least, Netflix works for search, but it doesn’t show up in the “continue watching” list. I don’t understand their motivation for that. And it means I often end up forgetting I’m watching a Netflix show because I don’t see “continue” until I end up back in Netflix.
On which one? On Amazon FireTV devices it does link into the Netflix app. It works well enough that to e.g. continue watching series on Netflix I rarely open the Netflix app because they're visible directly in the FireTV interface.
