I think maybe the problem with Shadow is that he's kind of a controversial character. He has a big following obviously (with good ol' Village and Brock leading the charge), but a lot of older Sonic fans and non-fans alike view him as the representation of Sonic's decline as a franchise. Maybe that influenced Sakurai to pass him on by.
Despite that, Shadow is one of the most popular characters after Sonic himself (I wanna say he's even the second most popular, before even Tails, Knuckles or Robotnik). In Sonic ensemble cast games, Shadow almost always is among the first characters revealed.
However, considering in Japan, Sonic always was more of a B-series, I think it's more that fact than how Shadow is perceived by some parts of a very split fanbase. Keep in mind that Sakurai does not seem to mind controversial picks - we got characters like Wii Fit Trainer or Corrin, which are characters that Sakurai most definitely anticipated negative backlash to, and we also got characters who definitely split the Smash fanbase, such as Ridley, as well.
If Sakurai had wanted to include Shadow, then I think we would have gotten him. In the same vein, even though characters like Ms. Pac-Man, Alph, Proto Man etc. would have made a lot of sense as additional Echoes too - and in the case of Alph, was planned to be such in Smash 4 - they ultimately were not chosen, and I think the reason for that is mainly because the Echoes we got in Ultimate, outside of Ken and Richter, all most definitely were big in the ballot.
Dark Samus and Chrom are basically confirmed ballot picks, and Daisy most likely is as well. Sakurai seemingly was intending to include Ken as clone in Smash 4 already, plus there's the whole "original Echo fighter" angle. Richter made sense to include because Simon's moveset already was heavily based off of Richter's abilities, and as Castlevania was the only new guest series, so it getting an Echo also wasn't completely out of the blue.
So, in the grand scheme of things, Shadow's omission from getting picked as an Echo makes sense and falls in line with the other options they would have had. I think however, if we had gotten like say 3 or 4 more Echoes, Shadow (or Metal Sonic) probably would have made the cut.
What would the closest thing to Japan's "Sonic" when it comes to Sega series. Would it be Puyo Puyo? (Not sure if Sega even created it?)
Puyo is Sega, yes, and also probably the biggest series from 'em in Japan.