In Targum Rishon of Esther 1:16, the character "Memuchan" is Haman the Agagite, whose name keeps reappearing from this point forward throughout the narrative (except in 1:12 where his name of "Memuchan" is mentioned yet one last time).
In Targum Rishon of Esther 4:5 "Hasach" appears, who is Daniel the prophet. The narrative continues to refer to him as "Hasach" several times, until in 4:12 he is killed by Haman, and the reader is once again explicitly reminded in the Targum that this person was in fact Daniel the prophet.
According to the Babylonian Talmud, the emphasis on Daniel's name "Hasach" stemmed for two reasons: (a) the name "Hasach" means cut down (killed by Haman); and (b), the name also means decided, which meant that Daniel's words (prophecy) had carried cosmic force.
b. Megillah (15a)
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The same commentary appears in the relevant Midrash.
Midrash Rabbah 8:4
“ ‘Then Esther called for Hasach.’ Our teachers there say that Hasach is the same as Daniel, and because he was cut down from his greatness they called him Hasach. Our teachers here say that he was so called because he made decisions on affairs of state.”
In summary, the emphasis on the name "Hasach" for Daniel in the narrative was to focus the reader on the humility ("cut down") and greatness of Daniel the prophet, whose words carried divine sanction and power ("he made decisions on affairs of state").
Thus it is that in 1:16, the narrative starts off with the name used in the Megillah and identifies who it is only after, while in 4:5, the narrative gives the character's actual name and then refers to him as the Megillah does.