Wasn't the Divine nature the soul/spirit of the body? In the same sense we incarnate, the soul/spirit is the part that matters, the Nous. Wasn't it the same with our Lord, with the exception of the soul/spirit/Nous being Christ?
No. The idea that Christ did not have a human soul/spirit is called Appollinarianism. As Deusveritasest said, His humanity was complete - He assumed all that we are, save sin.
So in this sense, is his humanity the same as that of the Theotokos?
As Salpy said above, yes, yes! To me personally, this is perhaps one of the most striking things of our faith. The Logos, God the Word, the second Person of the Most Holy Trinity, united in Himself the eternal, unchanging Divine nature with our human nature. The Fathers teach that the nature does not ever exist in an abstract sense - it is always "hypostatical," or, plainly, "someone's." So, the humanity, the human nature of the Logos did not exist until, at the moment of Annunciation, the "zygote" (human cell with its DNA, genes, program for the development of a future organism) in the absolutely human body of the Most Holy Theotokos started to grow into an embryo, and then fetus, and then baby, and then child, and then man Jesus Christ. His humanity was the product of Her humanity. He has Her genetic code and Her human traits: Her color and shape of the eyes, Her facial features and expressions, Her manner of speaking, posture, smile, etc. As God, He has no "structure," no "complexity" (Greek Απλους, "simple," in the sense, not consisting of anything but just being), no temporal and spacial limitations (at once everywhere, and ALL everywhere or anywhere), invisible, untouchable, incomprehensible. As man, however, He has our human body (even though "transfigured," with new properties that we don't have yet, like the ability to go through locked doors, or the ability to appear simultaneously in different places, etc.), and absolutely human features, human individual person, which He took from the human person of a humble Hebrew maiden, the daughter of Stt. Joakim and Anna! And when (and if...) we enter His eternal Kingdom after the resurrection, we will quite literally be sitting at His dining table, and He will put a towel around His waist, and wash our feet...