The Shaloh HaKadosh in Aseret HaDibrot, Yoma, Derekh Chayim famously writes that:
וחנוכה ופורים הם שני רעים - Chanukah and Purim are like two friends (my own translation)
What exactly does this Shaloh means?
Are there seforim that explain the meaning behind this Shaloh?
There are a possibly a few ways in which they can be deemed "friends", or in other words, share a similar connection.
The first, from a most basic level, is that they are both Rabbinic festivals that commemorate two clear miracles that result in us inserting al hanissim into the shemoneh esrei and bensching.
However, this is a deeper connection.
The Ramchal in Derech Hashem, Part 4 writes expressly:
חנוכה ופורים: ענין חנוכה ופורים הוא להאיר האור המאיר בימים ההם כפי התקונים שנתקנו בם. חנוכה בתגבורת הכהנים על הרשעים בני יון שהיו מתכונים להסיר ישראל מעבודת ה׳ ונתחזקו הכהנים ועל ידם שבו לתורה ולעבודה. ובפרט ענין המנורה לפי תיקוניה שהיו הקטרוגים נגד ענינה והחזירום הכהנים על בורים. ופורים לענין הצלתם של ישראל בגלות בבל וחזרת קבלת התורה שחזרו וקבלו עליהם לעולם כמו שאז״ל (שבת פח א) הדור קבלוה בימי אחשורוש ופרטי הענינם כפי פרטי התיקון
Chanukah and Purim: The content of Chanukah and Purim is to bring out the light that shines during those days, according to the refinements that were refined by them: Chanukah is about the priests (Kohanim) overpowering the evil Greeks, who intended to take Israel away from the service of God. But the priests strengthened themselves, and - through them - [the other Jews] came back to the Torah and the [divine] service, especially that of the Menorah according to its practices, about which there were accusers; whereas the priests brought it back to its propriety. And Purim concerns the matter of the salvation of Israel during the Babylonian exile; and the accepting of the Torah again. For they then accepted it again forever, as they, may their memory be blessed, said (Shabbat 88a), "They went back and accepted it at the time of Achashveirosh." And the details of their content are according to the details of the refinement.
The Sefas Emes you seek is here (The column on the left, second paragraph) with the key quote being:
כי חנוכה ופורים הם מחזיקים אותנו בגלות כי הם מאירים (בגלות) תוך החושך
Because Chanukah and Purim strengthen us - because they illuminate (in exile) in the darkness...
The Targum Simcha Here (English Here) mentions four similarities and connections between Purim and Chanukah.
Although he does not mention the Shelah which you quoted, he does provide his own connections between the two holidays.
The Lubavitcher Rebbe (Shabbos Parshas Mikeitz 27th Day of Kislev 5744) explained that Purim represents the body, e.g. mitzvos, and Chanukah represents the soul, e.g. Torah.
On Purim, we "accepted the Torah again". The Ohev Yisroel says:
But even so, they accepted it in the days of Achashveirosh, which was a time of great pain and distance from G-d, and even so, they accepted it upon their souls, even though they were in a very lowly state, and committed to do all that was written in the holy Torah. And so we find that there was no longer any mitigating claim about the acceptance of the Torah after they accepted it out of the desire of their hearts.
During Chanukkah, the Greeks forbade to study the Torah, especially the oral law, they forbade studying. But nevertheless, the light of Torah could not be diminished. Rabbi Pinchas Stopler (Hidden Lights: Chanukah and the Jewish/Greek Conflict) quotes Rav Hutner and says:
The determined decication of the Jews who confronted the Greeks in order to preserve the Torah initiated a fundamental change in the method of Torah transmission (p. 273 The Torah Became the Possession of Each Jew).
So maybe this is what Purim and Chanukah have in common, the renewal of Torah and the re-kindling of the "inner flame" of Creation.
