Racial Emulation nets you the subtype, though. That's what the feat does
Nope, please reread the Feat.
You emulate having the Subtype != You have the Subtype.
Kind of kicks you in the nuts before you even get into the material don't it?
Secondly the Feat outright states you do not gain any of the humanoid's traits.
EB:MoEp7 lists the Living Construct Subtype (including the construct type) as a Warforged Racial Trait.
EB:MoE25 lists the Shapechanger Subtype (including humanoid) as a Shifter Racial Trait.
EB:MoE41 lists the Shapechanger Subtype (including humanoid) as a Changling Racial Trait.
EB:MoE55 lists nothing for Kalashtar. Because all Humanoids(
racetype) forgo explicitly mentioning their Type/Subtype and assume you already understand such. Kind of an oversight on their end honestly, I mean PHB alone doesn't even explain why Favored Enemy works on Elves, you have to read both the Humanoid and Elf entries in the MM to even grasp the concept. But the fact of the matter is, Type/Subtype assignment is a Racial Trait.
You do not gain the Type/Subtype of the Race you are emulating and Racial Emulation never said you did to begin with. That's really all there is to it.