Maybe transmigrants and the originals don't always share the same power? Wasn't it implied that there where many other transmigrants other than nardo, sauce, hashi, and mads? They just didn't make the cut it seemed, if that's the case mokuton might be something hashi figured out on his own, just my own speculation
Almost all the transmigrants failed to awaken their power like the ones we've seen, if they unlocked their potential they would most likely have the same abilities as Sasuke and Hashi.
Only two transmigrants have awakened MS and one of them had Amaterasu which the manga implied Indra having. And Hashirama is the only known Mokuton user in Senju history, if he wasn't' it would've been mentioned.
In addition, it was shown that KKGs come from the chakra the person possesses, when a person possesses Hagoromo's chakra they get his eye abilities.
Likewise, when DMS Obito transmigrated into Kakashi we saw Kakashi being able to use Obito's abilities. The same logic should apply to Indra and Ashura transmigrating into their descendants
Hashirama's bloodline gave him Ashura's chakra which in turn would give him Mokuton while Naruto's compatibility with bijuus came from Ashura inheriting a body comparable to a JuubiJin. And Sasuke's eye abilities would come from having Indra's chakra.