I have never seen or even heard of this show. It sounds interesting, though stupid, for the reasons you mention.
A better match would be Spartan vs. samurai, as the ninja were spies and assassins, not warriors. They fought only when forced to.
A single first rate samurai would probably have taken out a top-notch Spartan in no time. The samurai were individualistic warriors. Spartan equipment and training was not based on individual combat, while that of the samurai was.
For the exact same reason, a thousand Spartans would have demolished a thousand samurai. The Spartans were trained to fight as a unit, and were punished and sometimes even executed for glory-hounding like the samurai routinely indulged in.
The show had a samurai vs. Viking and the samurai won because of his individual weapons and skill. As you mention, Vikings fought as a unit using their shields to build a wall.
I think your point is valid although obviously no one will ever know because we can’t have a real Spartan fight a real Samurai. I happened to see that episode (it is the only one I have seen) and their point about the Spartan using his shield as a weapon seemed like a good one.
Thucydides in his “History of the Peloponnesean War” mentioned that the Spartans were not extremely good as individual fighters but were unbeatable in the phalanx. I suspect they actually were pretty good as individuals since they spent their entire lives training.
My guess would be the same as yours. As individuals I would pick the Samurai, as an army the Spartans. It might vary more with individuals than anything else.
The Spartans main weapon was the dory or stabbing spear. It would probably be much less effective in individual combat but I personally would not want to face one. They of course also had their sword.