One of Rambam's arguments for the existence of only one God is that the existence of multiple gods by definition implies some form of corporeality.
Hilchos Yesodei HaTorah 1:7
אלוה זה אחד הוא ואינו שנים ולא יתר על שנים אלא אחד שאין כיחודו אחד מן האחדים הנמצאים בעולם לא אחד כמין שהוא כולל אחדים הרבה ולא אחד כגוף שהוא נחלק למחלקות ולקצוות אלא יחוד שאין יחוד אחר כמותו בעולם אילו היו אלוהות הרבה היו גופין וגויות מפני שאין הנמנים השוין במציאותן נפרדין זה מזה אלא במאורעין שיארעו בגופות והגויות ואילו היה היוצר גוף וגוייה היה לו קץ ותכלית שאי אפשר להיות גוף שאין לו קץ וכל שיש לגופו קץ ותכלית יש לכחו קץ וסוף ואלהינו ברוך שמו הואיל וכחו אין לו קץ ואינו פוסק שהרי הגלגל סובב תמיד אין כחו כח גוף והואיל ואינו גוף לא יארעו לו מאורעות הגופות כדי שיהא נחלק ונפרד מאחר לפיכך אי אפשר שיהיה אלא אחד וידיעת דבר זה מצות עשה שנאמר ה' אלהינו ה' אחד
This God is one. He is not two or more, but one, unified in a manner which [surpasses] any unity that is found in the world; i.e., He is not one in the manner of a general category which includes many individual entities, nor one in the way that the body is divided into different portions and dimensions. Rather, He is unified, and there exists no unity similar to His in this world.
If there were many gods1, they would have body and form, because like entities are separated from each other only through the circumstances associated with body and form.
Were the Creator to have body and form, He would have limitation and definition, because it is impossible for a body not to be limited. And any entity which itself is limited and defined [possesses] only limited and defined power. Since our God, blessed be His name, possesses unlimited power, as evidenced by the continuous revolution of the sphere, we see that His power is not the power of a body. Since He is not a body, the circumstances associated with bodies that produce division and separation are not relevant to Him. Therefore, it is impossible for Him to be anything other than one.
The knowledge of this concept fulfills a positive commandment, as [implied by Deuteronomy 6:4]: "[Hear, Israel,] God is our Lord, God is one." (Chabad.org)
However, when he discusses angels he asserts that they are also incorporeal, yet he states that there are many different angels.
Hilchos Yesodei HaTorah 2:2-5
כל מה שברא הקב"ה בעולמו נחלק לשלשה חלקים מהן ברואים שהן מחוברים מגולם וצורה והם הווים ונפסדים תמיד כמו גופות האדם והבהמה והצמחים והמתכות ומהן ברואים שהן מחוברים מגולם וצורה אבל אינן משתנין מגוף לגוף ומצורה לצורה כמו הראשונים אלא צורתן קבועה לעולם בגולמם ואינן משתנין כמו אלו והם הגלגלים והכוכבים שבהן ואין גולמם כשאר גולמים ולא צורתם כשאר צורות ומהן ברואים צורה בלא גולם כלל והם המלאכים שהמלאכים אינם גוף וגויה אלא צורות נפרדות זו מזו
ומהו זה שהנביאים אומרים שראו המלאך אש ובעל כנפים הכל במראה הנבואה ודרך חידה לומר שאינו גוף ואינו כבד כגופות הכבדים כמו שנאמר כי ה' אלהיך אש אוכלה הוא ואינו אש אלא משל וכמו שנאמר עושה מלאכיו רוחות
ובמה יפרדו הצורות זו מזו והרי אינן גופין לפי שאינן שוין במציאותן אלא כל אחד מהן למטה ממעלתו של חבירו והוא מצוי מכחו זה למעלה מזה והכל נמצאים מכחו של הקדוש ברוך הוא וטובו וזהו שרמז שלמה בחכמתו ואמר כי גבוה מעל גבוה שומר
Everything which the Holy One, blessed be He, created within His world is divided into three categories. They include:
a) Creations which are a combination of matter and form. They are constantly coming into existence and ceasing to exist; for example, the bodies of man and beasts, plants, and metals.
b) Creations which are [also] a combination of matter and form, but do not change from body to body and from form to form as those in the first category. Rather, their form is permanently fixed in their matter, and they do not change as the others do; for example, the spheres and the stars [which revolve] in them. The matter [from which] they [are composed] differs from [a simple conception of] matter, and their form differs from [a simple conception of] form.
c) Creations which have form, but no matter at all; for example, the angels, for the angels do not possess bodies or corporeal being, but rather are forms which are separate from each other.
What is meant by the prophets' statements that they saw an angel of fire or with wings? All these are prophetic visions and parables, as [Deuteronomy 4:24] states: "God, your Lord, is consuming fire," though He is not fire and [the description of Him in this manner] is only metaphoric. Similarly, [Psalms 104:4] states: "He makes His angels as winds...."
Since they possess no body, what separates the form [of the angels] from each other? Their existence is not alike. Rather each one is below the level of the other and exists by virtue of its influence, [in a progression of levels,] one above the other.
Everything exists by virtue of the influence of the Holy One, blessed be He, and His goodness. Solomon alluded to this [concept] in his wisdom, saying (Ecclesiastes 5:7): "Because above the one who is high there is a watcher [and there are others higher than them]." (Chabad.org)
Why is it that angels can be distinguished from each other without a physical manifestation, yet gods cannot be distinguished from each other without a physical manifestation?
Even if one were to argue, as does R. Elyahu Touger in his footnote to 1:7, that Rambam is even referring to "a spiritual sense of these concepts", it does not explain why gods cannot be differentiated while angels can.
If one were to point out that the very method of differentiation of angels is based on their hierarchy of levels which, if existing for gods would by definition render them non-gods (if there is a god on a higher level, the lower god by definition is lacking a quality of godliness) then Rambam's argument is really a different argument – that the existence of multiple gods is impossible because the definition of god entails absolute power and it is impossible for two separate entities to both have absolute power (as discussed by Saadia Gaon in Emunos V'Deios 2:2).
See also Guide for the Perplexed 2:1 where Rambam writes:
The hypothesis that there exist two Gods is inadmissible, because absolutely incorporeal beings cannot be counted (Prop. XVI.), except as cause and effect; the relation of time is not applicable to God (Prop. XV.), because motion cannot be predicated of Him. (Friedlander translation.)
Once again it appears that this argument must necessarily be predicated on the argument of impossibility of two absolute powers. Thus, the argument from the impossibility of distinguishing incorporeal entities should either be insufficient or unnecessary to prove that there is only one god.
Any answers, or flaws/incorrect assumptions in the question?
[I think the Peirush sort of addresses this, but he makes the proof dependent on a different earlier premise, namely, that there can only be one First Cause. If that is the case then once again this specific proof should be either insufficient or unnecessary.]
1.One of the answers challenged this translation, and said that the word here is "godliness" rather than gods. I would therefore note that the Feldheim translation by Moses Hyamson agrees with R. Touger's translation:
If there were plural deities, these would be physical bodies;