Cloud and Link's pop culture rivalry emerged a lot more organically than the Mario vs. Sonic or Mario vs. Crash rivalry did.
The latter examples did catch on with the general public but were intentionally invented and (in many ways) furthered by marking teams and developers. Link vs. Cloud is really just a thing because Ocarina of Time and Final Fantasy VII had the coincidental fortune of both being major, groundbreaking Japanese games that used a Western fantasy setting (although in FF7's case, there's a lot of Western Sci-fi as well) and featured a young, blonde man as their protagonist that released in the late 90s. Being exclusives on the two best-selling consoles in the Western market also helped.
Neither Square nor Nintendo saw their respective product or character as the answer to the other's, but the overlap in content and the fact that they were leading examples of what direction games were going in on two products that did have a public rivalry led to consumers making an association between the two games and, by extension, their leading characters.
By 2015, when Cloud was finally added to Smash and the two characters could directly compete in a product officially sanctioned by both their parent companies, the idea of Link vs. Cloud had grown from a grassroots playground debate to a genuine social phenomenon universally recognized by those interested in gaming or pop culture as a whole. As such, it's unsurprising that Nomura would make Link vs. Cloud the centrepiece of his art for Cloud's inclusion in Smash for Wii U/3DS.