Moshiach is an individual who is present in the world during the time of exile and in the location of the exile. He is in an “exiled” state of being, as he himself suffers from the afflictions of the exile.

…מָשִׁיחַ נִמְצָא בָּעוֹלָם בִּזְמַן וּמְקוֹם הַגָּלוּת, וּבְמַצָּב שֶׁל גָּלוּת, שֶׁסּוֹבֵל תַּחֲלוּאֵי הַגָּלוּת

This is evident from what the Gemara states in continuation of the topic of Moshiach’s name: “Rav said: if he, Moshiach, is from the living, he is like our holy Rebbi Yehudah Hanasi.”

[כְּהֶמְשֵׁךְ הַגְּמָרָא “אָמַר רַב אִי מִן חִיָּא הוּא כְּגוֹן רַבֵּינוּ הַקָּדוֹשׁ”,

Rashi explains why Rebbi, out of the many great sages of the time, was cited as an example for the Moshiach of the generation:

“If Moshiach is from those that are living now, he is certainly our holy Rebbi, who bore afflictions and was completely pious.”

“אִם מָשִׁיחַ מֵאוֹתָן שֶׁחַיִּן עַכְשָׁו וַדַּאי הַיְנוּ רַבֵּינוּ הַקָּדוֹשׁ, דְּסוֹבֵל תַּחֲלוּאִים וְחָסִיד גָּמוּר הַוֵה”],

In other words, besides for Moshiach obviously being a pious and righteous man, he is also identified by being someone who suffers the pains of exile.

Furthermore, he waits impatiently and longingly to be revealed from his concealment during the exile

וּמְצַפֶּה בְּקֹצֶר רוּחַ וּבְכִלְיוֹן עֵינַיִם לְהִתְגַּלוֹת (מֵהַהֶעֱלֵם בִּזְמַן וּמַצָּב הַגָּלוּת),

as the King Moshiach who redeems the Jewish people at the true and complete Redemption, titled so because after it there will be no exile,

דְּמֶלֶךְ הַמָּשִׁיחַ גּוֹאֵל אֶת יִשְׂרָאֵל בַּגְאוּלָה הָאֲמִתִּית וְהַשְׁלֵימָה שֶׁאֵין אַחֲרֶיהָ גָּלוּת,

following the halachic ruling of Rambam, that ultimately “A king will arise from the house of David” who will bring about the ultimate Redemption.

כִּפְסָק-דִּין הָרַמְבָּ”ם “יַעֲמֹד מֶלֶךְ מִבֵּית דָּוִד וְכוּ’”.

To further support this concept, the Rebbe cites in a footnote how various schools of sages in the time of the Mishnah would point to their rabbi as a possible redeemer, indicating that Moshiach is a figure found in exile:

There, at the beginning of the discussion in the Gemara we find thus:

… וְשָׁם (בְּהַתְחָלַת הַסּוּגְיָא):

The school of Rabbi Sheila says: Shiloh is the redeemer’s name, as it is stated: “Leadership shall not depart from the tribe of Yehudah until when Shiloh shall come;”

“דְּבֵי רַבִּי שֵׁילָא אַמְרֵי שִׁילֹה שְׁמוֹ, שֶׁנֶּאֱמַר עַד כִּי יָבוֹא שִׁילֹה,

The school of Rabbi Yannai says: Yinnon — similar to “Yannai” — is his name, as it is stated concerning Moshiach’s ultimate greatness: “May his name endure forever; may his name continue [yinnon] as long as the sun;”

 דְּבֵי רַבִּי יַנַּאי אַמְרֵי יִנוֹן שְׁמוֹ .. שֶׁנֶּאֱמַר יְהִי שְׁמוֹ לְעוֹלָם לִפְנֵי שֶׁמֶשׁ יִנוֹן שְׁמוֹ,

(Rashi explains that “each one derived a name for Moshiach from the scriptures, modeled after his name.”)

(כְּמוֹ יַנַּאי, כָּל אֶחָד הָיָה דּוֹרֵשׁ אַחַר שְׁמוֹ),

[Rashi’s wording is precise: “each one derived … his name,” and not that the students introduced the idea that his [Moshiach’s] name is like that of their rabbi.]

[וּלְהָעִיר מִדִּיּוּק לְשׁוֹן רַשִׁ”י “כָּל אֶחָד הָיָה דּוֹרֵשׁ אַחֵר שְׁמוֹ” (וְלֹא שֶׁהַתַּלְמִידִים חִדְּשׁוּ שֶׁשְּׁמוֹ כְּשֵׁם רַבָּם).]

… And some say that Menachem ben Chizkiya is his name, as it is stated: “Because the comforter [menachem] that should relieve my soul is far from me.”

… וְיֵשׁ אוֹמְרִים מְנַחֵם בֶּן חִזְקִיָּה שְׁמוֹ, שֶׁנֶּאֱמַר כִּי רָחַק מִמֶּנִּי מְנַחֵם מֵשִׁיב נַפְשִׁי”.

And we, as Chassidim, should follow their example in regard to identifying our Rebbeim and leaders as Moshiach, and the Rebbe and leader of our generation, my saintly father-in-law in particular:

וַאֲנָן (חֲסִידִים) נַעֲנֵיהּ אַבַּתְרַיְיהוּ (בְּנוֹגֵעַ לְרַבּוֹתֵינוּ נְשִׂיאינוּ, וּבִפְרָט כְּבוֹד-קְדֻשַׁת מוֹרִי-וְחָמִי אַדְמוֹ”ר נְשִׂיא דוֹרֵנוּ) –

Yosef is his name, as it is stated “The L-rd will continue [yosif] to apply His hand a second time … and gather the lost of Israel…” Yitzchak is his name, as it is stated, “Then, our mouths will be filled with laughter [sechok].”

יוֹסֵף שְׁמוֹ, שֶׁנֶּאֱמַר “יוֹסִיף אַדֹנָ-י שֵׁנִית יָדוֹ גוֹ’ וְאָסַף נִדְחֵי יִשְׂרָאֵל גוֹ’”, יִצְחָק שְׁמוֹ, שֶׁנֶּאֱמַר “אָז יְמָלֵּא שְׂחֹק פִּינוּ”.