The Zohar points out (Vayechi 238b) that many mitzvos work in pairs, or have at least two components. For example, Tefillin & Tzitzis, Milah & Pri’ah, Mezuzah & Chanukah lights. Where there’s one, you’ll find the other. This is because creation itself is essentially a dichotomy that breaks down even further, but always stems from that first act of separation (from our perspective) from G-d’s ultimate unity; as the Torah teaches: “In the beginning G-d created the heavens and the earth”: Heaven and earth, Chesed and overabundant bestowal along with Din and receptive limitation. And with this come the dual forces of Mashiach ben Yosef and Mashiach ben David, the two components of Mashiach, working to perfect the union of heaven and earth.
On Succos, a holiday filled with mitzvos, there are the two that stand out the most: Lulav & Succah. If we take the Kabbalistic dynamic of Yesod & Malchus, which, in short, complete the relationship of the sefiros, Lulav is connected most strongly to the aspect of Yesod, while Succah is connected to Malchus. Since we are judged on Succos for the heavenly bestowal of rain (Rosh Hashanah 1:2), we take the Lulav, all of whose minim rely on rain, bind and take them together, to create the influx of divine energy that corresponds to rain. Since Yesod is essentially a channeling sefirah, which binds together the sefiros and channels them into Malchus, the Lulav (which of minim itself corresponds to Yesod and being the dominant quality, the whole group is called Lulav-Yesod) corresponds to Mashiach ben Yosef as he is manifest as Yesod, the spiritual potential which becomes actualized in Malchus, namely, Mashiach ben David.
The Succah is even called “Succas David HaNofales”, the Malchus of David waiting to be restored and re-established from its fallen state since the exile.
Kabbalah often calls Malchus a house, because it houses and receives the upper sefiros, and like the home, is the actualization of all of a person’s spiritual and material potential’ and the Succah is supposed to be our actual house for the duration of Succos. We also find that Malchus is the dominant sefirah by Succah (in a similar way that Lulav is dominantly Yesod) by the halacha that a Succah can minimally be 7 by 7 tefachim, seven corresponding to the 7th sefirah, Malchus containing the light of seven sefiros, 7 by 7.
Perhaps this is part of why the AriZaL stressed the practice of first taking the Lulav in the Succah itself and blessing on it there. The point is to manifest that unity of Yesod & Malchus, bringing the Lulav-Yesod, into Succah-Malchus, for if Succah is the vessel to receive and actualize, the Lulav, which is bestowal of divine energy and material in potential form, should be taken there, to close the circuit. With this practice, Heaven-the Lulav unites with the earth-Succah.
R’ Nosson teaches in the name of R’ Nachman (in Likkutei Halachos, Succos) that the Succah is the holy spiritual container and boundary in which it is safe to bring in the material and potential aspects of life and elevate them. That’s also part of why we live, eat and sleep there. The Succah is kind of “hot box” of holiness, where physicality can be safely drawn in and made holy.
This is part of the reason why the Torah tells us that Yaakov “made Succos for his animals”, because we are meant to bring the material and mundane into the framework of holiness, as long as that framework has been properly established.
This is also why the nations are strongly associated with the holiday of Succos because they too are to be elevated within the grand spiritual framework of the Succah (which may also be why the account Yaakov making succos for his animals follows Yaakov’s promise of future union with Eisav, for Yaakov is making the preparations through Succah to achieve the proper relationship to Eisav, the nations).
This is all hinted at in the gematria of Succah being equal to that of G-d’s Names, Adnus & Yud Kei Vav Kei, which are chesed and din, Yesod and Malchus, Mashiach be Yosef and Mashiach ben David.