Question:

Was Haman the Agagite (Esther 3:1)an Amalekite (1 Samuel 15:32) or is it just a coincidence?

Answer:

I will admit that I am shamelessly copying and pasting from a Jewish web site, as I probably cannot improve on their knowledge.  Apparently, Josephus agreed that Haman was an Amalekite.  The veracity of some Jewish legends about Haman is dubious, but that Haman was an Amalekite seems, to me, to be fairly solid.

John O.

From:  http://www.jewishencyclopedia.com/articles/7124-haman-the-agagite

Son of Hammedatha; chief minister of King Ahasuerus (Esth.iii.1-2). As his name indicates, Haman was a descendant of Agag, the king of the Amalekites. On account of his attempt to exterminate the Jews in the kingdom of Ahasuerus, he is frequently called “the persecutor of the Jews” ( ; Esth. iii. 10; viii. 1; ix. 10, 24). His machinations against the Jews and his downfall are remembered during the Feast of Purim. Filled with annoyance because Mordecai did not bow before him, Haman resolved upon the extermination of the Jews throughout the whole kingdom. He drew lots to determine the day of the massacre, and the lot fell on the 13th of Adar (Esth. iii. 4-7). He offered the king ten thousand talents of silver for permission to do with the Jews as he pleased. The permission was granted, and he accordingly despatched letters to all parts of the Persian kingdom to massacre the Jews on the 13th of Adar (iii. 8-15). His intrigues, however, were baffled by Esther. In order to throw him off his guard she invited him to a banquet to which she had also asked the king. Haman, looking upon this as an indication of special favor, in his pride went so far as to prepare a gallows whereon to hang Mordecai (v. 14). But in that night a sudden change occurred in Haman’s fortunes. His own answer to the king’s question what should be done to him whom the king delighted to honor, which Haman supposed referred to himself, obliged Haman to lead Mordecai, his mortal enemy, clad in royal garments and seated on the king’s horse, through the streets of Shushan and to proclaim: “Thus shall it be done to the man whom the king delighteth to honor” (vi. 9). Afterward, while Haman was again drinking with the king at a banquet prepared by Esther, the latter exposed to the king Haman’s plot. The king, filled with anger, ordered his officers to hang Haman on the very gallows which he had prepared for Mordecai (vii. 9). Ahasuerus bestowed upon Esther Haman’s house (viii. 1); the ten sons of Haman were executed on the 13th of Adar and then hanged (ix. 7-9, 14).

—In Rabbinical Literature:

Haman is identified by the Talmudists with Memucan, the last of the seven princes “which saw the king’s face” (Esth. i. 14), giving to “Memucan” the signification of “prepared for punishment” (Targ. to Esth.; Meg. 12b). Haman was a direct descendant of Agag in the sixteenth generation and consequently an Amalekite (Targ. Sheni; Josephus, “Ant.” xi. 6, § 5). The Septuagint, however, gives for “ha-Agagi” ὅ Μακεδόν in Esth. ix. 24, while in the preceding instances no translation whatever is given. Having attempted to exterminate the Jews of Persia, and rendering himself thereby their worst enemy, Haman naturally became the center of many Talmudic legends. Being at one time in extreme want, he sold himself as a slave to Mordecai (Meg. 15a). He was a barber at Kefar Ḳarẓum for the space of twenty-two years (ib. 16a). Haman had an idolatrous image embroidered on his garments, so that those who bowed to him at command of the king bowed also to the image (Esth. R. vii.).