I'd like to start by saying Dr. Wily plagiarises MarioWiki and so I didn't watch the video. (Lol, I'm listening to the MM7 soundtrack rn and just got the the Wily Castle songs, what a coincidence, also those songs are the best in the game, but anyways...)
The answer is yes... kinda. I mean, we see the two universes collide on various occasions outside of Smash (which is about a kid playing with toys, so it doesn't count in any discussion outside of very rare circumstances). Link's Awakening features various Mario characters, but that was all a dream... and so was Super Mario Bros. 2, so perhaps Koholint and Subcon are in some kind of connected dream dimension? Makes sense to me. The Mario series has never been very consistent with it's canon (which is why MarioWiki abandoned the concept of canon entirely), so perhaps Club Nintendo and similar occurrences wouldn't count in this discussion, but as mαrkαsscoρ
said, Link appears in SMRPG, and he also appears in the Donkey Kong Country games. We also saw Link in Mario Kart, but that has its own discrepancies with the canon, with Baby Mario and Metal Mario and Pink Gold Peach and Baby Rosalina and so on, so forth. There's also Mario's various cameos in ALttP and OoT as a painting, but since those are set "three or four generations" apart, I don't see how there could be any connection, unless Mario has a time machine... oh wait...
I think connecting the Nintendo universes is a really interesting concept. I really like how minor references can sometimes imply some big connected universe sometimes. Like with Link and Samus in SMRPG, Samus in Kirby's Dream Land 3, Mario and other Nintendo stuff in Kirby Super Star, et cetera. But really, I don't care either way.