It talks about the capital and "the king's house", but never uses the Hebrew word for "palace", "armon", as appears in Psalms and elsewhere. (Or does it appear and I missed it?) Why not?
Megillat Esther is from Ketuvim, which means written with Divine Inspiration, rather than with nevuah as dictation from Hashem. Even within Neviim, different neviim express themselves differently. If different words were common in different places and different times, it is not surprising that different phraseology might be used.
In Megillah 5:1, we read:
וַיְהִי בַּיּוֹם הַשְּׁלִישִׁי, וַתִּלְבַּשׁ אֶסְתֵּר מַלְכוּת, וַתַּעֲמֹד בַּחֲצַר בֵּית-הַמֶּלֶךְ הַפְּנִימִית, נֹכַח בֵּית הַמֶּלֶךְ; וְהַמֶּלֶךְ יוֹשֵׁב עַל-כִּסֵּא מַלְכוּתוֹ, בְּבֵית הַמַּלְכוּת, נֹכַח, פֶּתַח הַבָּיִת
"Now it came to pass on the third day, that Esther put on her royal apparel, and stood in the inner court of the king's house, over against the king's house; and the king sat upon his royal throne in the royal house, over against the entrance of the house."
Given that the king is sitting on his royal throne in his royal house, it would seem that בֵּית-הַמֶּלֶךְ and בֵית הַמַּלְכוּת refer to the palace.
My guess is the whole book's style is really centered on the ego of King Achashverosh. So it's always "the king's this", "the king's that." Similarly, almost all the time the king's workers do something it's passive voice "the matter was investigated" ... "the chronicles were read" ... the employees themselves don't exist!
Perhaps with the hidden sarcasm that this hadn't been the palace of Persia very long; the king was actually a stableboy who worked his way to the top. But that's a random thought ...