So in Wicked, we know that Elphaba puts a spell on Fyiero so he wouldn't get hurt, making Fyiero turn into the Scarecrow.
While in The Wizard of Oz, Elphaba is literally seen by trying to harm Scarecrow.
And at the end of Wicked, Scarecrow/Fyiero finds Elphaba and they leave Oz together.
This lead me to to 2 things:
Elphaba knows that Scarecrow is Fyiero, and trys to harm him just to see if her spell actually worked. And Scarecrow is scared of Elphaba/ doesn't who Elphaba is, due to not having any brains. So when Scarecrow has his brains back, he remembers Elphaba, and they run away together.
Elphaba knows that Scarecrow is Fyiero this time, but Scarecrow knows who's Elphaba all this time, and tricks everyone into believing that they don't know each other (but they do, and run away together)
So that's my thoughts on Scarecrow and Elphaba during TWoO. But I think number 1 makes more sense.
Videos
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(Book) Why did Elphie think Fiyero was The Scarecrow?
The Scarecrow's Identity: Wouldn't They Recognize Him in part 2?
Is Fiyero was the scarecrow, why did he join forces with Dorothy?
So Fiyero in the musical does in fact end up being The Scarecrow. But what led to Elphie thinking he was The Scarecrow in the book? Maybe I'm missing something, but the line of reasoning seemed to come out of nowhere. I've yet to read the sequels.
In "Wicked" Part 2, Jonathan Bailey turns into the Scarecrow, and even after his transformation, he still wears the same Captain's uniform and accompanies Dorothy to meet Oz. My biggest question is: Fiyero was the Captain of the Guard, so wouldn't Oz, Galinda, or Madame Morrible recognize his outfit and figure out his identity???