How do Orthodox Christians explain that the Father and the Son are not two different Gods?

The two Creeds are not different in respect of there being two/three Gods, because there is only one God, period. What's different is the "emphasis" of the two Creeds. The following explains it.

"The chief difference between the Nicene Creed and the Athanasian Creed is one of emphasis. The Nicene Creed emphasized the full deity of the Son and of the Holy Ghost and it only implies, rather than explicitly explains the oneness of God. The Athanasian Creed, leaving nothing to be assumed, brings in the idea of the inviolate Oneness of God (there is only one God, not three.) 8. The Athanasian Creed also deals with one additional error. At the time there were controversies being raised by some about the person and nature of Jesus Christ and whether or not He was both true God and true man. All of the rationalizations, in one way or another, either removed from Jesus his true humanity or his true divinity or the idea that there was just one Jesus who was both true God and true man. 9. The Church Year: Early Christians took the incarnation very seriously: They expected to follow the life example of Christ. They followed the church calendar through the various segments of Christ’s life. In the Western Churches, the Church Year was and remains to this day, divided into two almost-equal parts: the first some have called “The Life of Christ” while the second is “The Christian Life.” 10. The Year begins with Advent – a time of reflection in anticipation of the coming of Christ. Advent is followed by Epiphany, celebrating the revealing of Christ as God’s anointed one." https://www.coursehero.com/file/p5g865/The-chief-difference-between-the-Nicene-Creed-and-the-Athanasian-Creed-is-one/

You might find the following article informative as it relates to the "Patristic" fathers. https://www.monergism.com/thethreshold/articles/onsite/qna/patristics.html

The Father is θεος in a unique sense. He possesses qualities or traits that are unique to him alone:

(a) absolute authority - everything else came from the Father (the Monarchy of the Father)

(b) aseity - the Father came from no one else.

These ideas were found in 1 Cor 8:6, 15:26-27

The Father as the only true God is the sole source of both the Godhead and of the Trinity. Thus, when Holy Scripture calls him the only God or the one God, it means that the Father has a superior status, not superior ontology. The Father possesses the single Godhead (divine nature) from himself for no one gave it to him but that the Godhead just is the Father. The Father being the origin of the Son and the Holy Spirit is superior to them in function because (a) they came from him and (b) the divine essence they possess is his own essence. In this case, it logically follows that since they share the same numerical essence (ουσια) with him, they are not different Gods but one God (by nature) with him. (Note: θεος here is qualitative in sense as in John 1:1c "θεος ην ο λογος", so that one God = one ontological God, or one Divine Being).

For the Jewish background of the identity of the Father as the only God/only true God: In Hebraic terms, the Father would be Elyon (Most High God) among Elohim (gods), as found in Psalm 82:1, 6. These gods were actually his "sons" (Ben Elohim) as also found in the same text (Psalm 82:6). These sons were called gods because they participate in the divine nature (φυρις) of their father who is God (cf. 2 Peter 1:4). They are gods by nature (as opposed to idols who were by nature ( φυσις) no gods - Galatians 4:8). However, they do not possess all the fullness of deity (πλερομα θεοτης) but only share in some of his divine attributes. The existence of other gods did not diminish the singular identity of Yahweh because he, as the most high god, is functionally superior over the other gods. This concept is present in the New Testament (compare the only god of John 17:3 with the gods in John 10:35). In another question (https://christianity.stackexchange.com/a/72155/5256), I explained John 17:3 exegetically under ancient Jewish concept of the divine council.

Note: Christians also will become gods via theosis which is the Orthodox teaching on salvation (soteriology) wherein being saved is a lifelong process which involves metaphysical changes in both body and soul, transforming it in the likeness of the crucified Christ (https://en.m.wikipedia.org/wiki/Theosis_(Eastern_Christian_theology)

Jesus Christ as the only begotten (μονογενης) of the Father, share in the same being (ομοουσιος) (Nicene Creed) with the Father. That is, they are not different Gods because they share in the same numerical essence. (Latin: essentia - being).

The divine nature of the Son was neither begotten nor created but was identical or the same as the Father's. They are of the same nature (ομοουσιος). They are of the same species, so to speak. In Aristotle, this refers to secondary ουσια (the universal or common nature, e.g. h*m* sapiens sapiens, one human nature common to all men). Now there also exists a primary ουσια which is the thing (subsistence) in which nature or species (substance) is predicated (e.g. an individual man or a human being, a horse etc.). Since the Father is the one God, then, it follows that the Father is the one (divine) being and since the Son came from the being of the Father ("ἐκ τῆς οὐσίας τοῦ Πατρος", a phrase found only in the original Nicene Creed of 325 C.E.), the Son is of the same being (again, ομοουσιος) with the Father.

And in one Lord, Jesus Christ, the Son of God, begotten from the Father, only-begotten, That is, from the substance of the Father (ἐκ τῆς οὐσίας τοῦ Πατρος) God from God

A one human being is one self (one human person) but the one divine being is not unitarian (one divine person) but trinitarian (three divine persons) since both the Son and the Holy Spirit came from the Father (i.e. they came from within the one being of the Father and hence, are inseparable from the Father). The Son and Holy Spirit came from the Father via eternal begetting and eternal procession, respectively, which were necessary actions within the being of the Father. The Son and the Holy Spirit were inseparable from the being of the Father and necessary to the Father's existence in the sense that the Father, being the only wise God, cannot exist without Wisdom, or being the Glory, cannot exist without his radiance (cf. Hebrews 1:3). They are not three independent divine beings (three Gods). They are literally one independent existence ( = one being). They are one being (a) as to their activity for they have one action (i.e. they do the same activity) and (b) as to their nature (i.e. they possess the same attributes of eternality, omnipotence etc.). In the one being called 'God' exists three divine persons: Father, Son and Holy Spirit. The Son and the Spirit specifically came out from (εκ) that one being identified with the first divine person. Thus, they are not separate Gods but one God, because they came from the Father, who is the one God and they are inseparable from him.

"It is absolutely necessary for us to be guided to the investigation of the Divine nature by its operations. If, then, we see that the operations which are wrought by the Father and the Son and the Holy Spirit differ one from the other, we shall conjecture from the different character of the operations that the natures which operate are also different. For it cannot be that things which differ in their very nature should agree in the form of their operation: fire does not chill, nor ice give warmth, but their operations are distinguished together with the difference between their natures. If, on the other hand, we understand that the operation of the Father, the Son, and the Holy Spirit is one, differing or varying in nothing, the oneness of their nature must needs be inferred from the identity of their operation" (On the Holy Trinity, Gregory of Nyssa)

"For the Word, being Son of the One God, is referred to Him of whom also He is; so that Father and Son are two, yet the Monad of the Godhead is indivisible and inseparable. And thus too we preserve One Beginning of Godhead and not two Beginnings, whence there is strictly a Monarchy. And of this very Beginning the Word is by nature Son, not as if another beginning, subsisting by Himself, nor having come into being externally to that Beginning, lest from that diversity a Dyarchy and Polyarchy should ensue; but of the one Beginning He is own Son, own Wisdom, own Word, existing from It. For, according to John, 'in' that 'Beginning was the Word, and the Word was with God,' for the Beginning was God; and since He is from It, therefore also 'the Word was God.' And as there is one Beginning and therefore one God, so one is that Essence and Subsistence which indeed and truly and really is, and which said 'I am that I am Exodus 3:14,' and not two, that there be not two Beginnings; and from the One, a Son in nature and truth, is Its own Word, Its Wisdom, Its Power, and inseparable from It. And as there is not another essence, lest there be two Beginnings, so the Word which is from that One Essence has no dissolution, nor is a sound significative, but is an essential Word and essential Wisdom, which is the true Son." (Discourse Against the Arians 4, Athanasius)

The one Godhead (divine nature) by which the Father is known as one God in an ontological sense (the Deity - θεοτης) is exactly the same Godhead subsisting in the Son and the Holy Spirit. The Father possesses the divine nature from himself so that he is one God in a functional sense (i.e. as the only divine person who is functionally superior, supreme in role e.g. as the origin of everything else that exists whilst he himself has no origin but possesses aseity) whilst the Son and the Holy Spirit possess the divine nature from the Father so that they are one God in ontological sense with the Father. In this case, the divine nature is numerically the same for each divine person. They are equal in all things except in their existence as three distinct divine persons under the names Father, Son and Holy Spirit (cf. Matthew 28:19).

The Father is therefore called θεος in a peculiar sense (as the most high God which is his ancient Jewish appellation, signifying his superior authority) which only carries functional subordination in relation to other persons who are called θεος. Creatures possess some of the divine attributes whilst the Trinity is consubstantial (possessing all the divine attributes).

The Son and the Holy Spirit possess the divine nature in the same way the true God himself (i.e. the Father) possesses it i.e. eternally and subsisting in only one place, at the Divine Being (προς τον θεον) or the Divinity (το θειον) which is identical to the Father. In this case, they are literally of the same being (ομοουσιος). Thus, The Trinity is deemed as monotheistic due to the person of the Father in Orthodox Christianity.