Here's a random thought:

What if Sans is the only one who can dodge because he's the only one who needs to dodge?

Think about it.

Even the weakest monsters in the Ruins or Asgore himself don't dodge, not even Undyne, why?

Because they have lots of health?

They think that no matter how hard the human, or you in this case, try your blows won't kill them. They will always have health left over.

And Sans is the only one who can dodge because he only has 1HP, meaning that any well planned attack can kill him, he could turn to dust at any moment, by anyone's hands, or even a simple trip or hit to the head, so he has to be on his toes when there's conflict or in general.

'Why doesn't he just teach the others how to dodge?', you may ask, well he may see it as a waste of time. No matter how many times he may teach others, the timeline might reset and revert everyone to their default fighting actions, discarding the ability to dodge altogether.

Well that's what I think.

There's a theory out there of how the skelebros could be the manifestation of Gaster's split soul after colliding with a human (soul) when he fell into the CORE (somefuckinghow)

And there's another one where Gaster is their actual dad

So I'm going to tackle both on this same topic.

In dealing with the first theory, what if Gaster, in his time, was the only one who could dodge but since he was a scientist and not a fighter, no one saw him used that skill or he was never in a fight to use it? And when his soul split that ability went to Sans but not Papyrus. The Bros are complete opposites so it's possible for one to have something the other doesn't.

With the second theory now, what if the ability to dodge is inherited rather than acquired? Like, again, Gaster could dodge and so that trait got passed down to Sans but for some reason not Papyrus, or he could be a late bloomer. And since both Sans and his father could dodge, Gaster trained him to use that skill when he wasn't working.

These three theories proclaim that dodging could be inherited, learned, taught, but to a select few.

Hell, we could even go to say that skeleton monsters are the only ones who can dodge no matter how many times its tried to be taught or learned. Which again would say that Papyrus either hasn't acquired that skill or he can't altogether.