I will only quote works I have read although a quotemine is possible, so that there is no confusion:
St. Gregory Nazianzen in the Third Theological Oration is most clearly using "generation" in the sense of "offspring," for he says:
"When did these [the Three Persons] come into being? They are above all 'When.' But, if I am to speak with something more of boldness — when the Father did. And when did the Father come into being. There never was a time when He was not. And the same thing is true of the Son and the Holy Ghost. Ask me again, and again I will answer you, When was the Son begotten? When the Father was not begotten. And when did the Holy Ghost proceed? When the Son was, not proceeding but, begotten — beyond the sphere of time, and above the grasp of reason; although we cannot set forth that which is above time, if we avoid as we desire any expression which conveys the idea of time.
"How then are They not alike unoriginate, if They are coeternal? Because They are from Him, though not after Him. For that which is unoriginate is eternal, but that which is eternal is not necessarily unoriginate, so long as it may be referred to the Father as its origin. Therefore in respect of Cause They are not unoriginate; but it is evident that the Cause is not necessarily prior to its effects, for the sun is not prior to its light.
"But how can this generation be passionless? In that it is incorporeal. For if corporeal generation involves passion, incorporeal generation excludes it... Nay, I marvel that you do not venture so far as to conceive of marriages and times of pregnancy, and dangers of miscarriage, as if the Father could not have begotten at all if He had not begotten thus; or again, that you did not count up the modes of generation of birds and beasts and fishes, and bring under some one of them the Divine and Ineffable Generation, or even eliminate the Son out of your new hypothesis. And you cannot even see this, that as His Generation according to the flesh differs from all others (for where among men do you know of a Virgin Mother?), so does He differ also in His spiritual Generation; or rather He, Whose Existence is not the same as ours, differs from us also in His Generation.
"How then was He begotten? This Generation would have been no great thing, if you could have comprehended it who have no real knowledge even of your own generation, or at least who comprehend very little of it, and of that little you are ashamed to speak; and then do you think you know the whole?... How was He begotten?— I repeat the question in indignation. The Begetting of God must be honoured by silence. It is a great thing for you to learn that He was begotten. But the manner of His generation we will not admit that even Angels can conceive, much less you. Shall I tell you how it was? It was in a manner known to the Father Who begot, and to the Son Who was begotten. Anything more than this is hidden by a cloud, and escapes your dim sight."
Notice to what He compares and contrasts the Eternal Generation of the Son, the Saint is not using the word in the sense of "unique," and also refutes that the Son being born of the Father requires that the Son did not exist at some point in time. He does not at all mean "unique." And again I say: the Creed recited at Church weekly says "born" after "only-begotten."
As for St. Athanasius in the very first discourse against the Arians (the above quotes by St. Gregory are against the Eunomians, a particular kind of Arian) he says, in the very first chapters of this work:
"Very Son of the Father, natural and genuine, proper to His essence, Wisdom Only-begotten, and Very and Only Word of God is He; not a creature or work, but an offspring proper to the Father's essence. Wherefore He is very God, existing one in essence with the very Father; while other beings, to whom He said, 'I said you are Gods ,' had this grace from the Father, only by participation of the Word, through the Spirit.
"For the Father and the Son were not generated from some pre-existing origin, that we may account Them brothers, but the Father is the Origin of the Son and begot Him; and the Father is Father, and not born the Son of any; and the Son is Son, and not brother. Further, if He is called the eternal offspring of the Father, He is rightly so called. For never was the essence of the Father imperfect, that what is proper to it should be added afterwards; nor, as man from man, has the Son been begotten, so as to be later than His Father's existence, but He is God's offspring, and as being proper Son of God, who is ever, He exists eternally. For, whereas it is proper to men to beget in time, from the imperfection of their nature, God's offspring is eternal, for His nature is ever perfect."
The Saint does not mean "unique" here also. He means precisely what he says, generation, offspring, and notice the contrasts he draws. And again I say: the Creed says "born."
Now is the Son unique? Obviously so, God does not relativize His Son (borrowing a saying from Fr. Staniloae), He is truly unique, but He is also Generated, Born, Birthed, from the Father before all ages. The doctrine is literally called the Eternal Generation. The Son was "born from the Father before all ages and born from the Virgin Mary in time" as is commonly said, I believe some other Fathers have said that and I remember one saying "eternal birth" but I do not remember that work title and may have not read it in full so I'll not try to quote it.
If you want to read these works I have given you their titles, you will see that I am not lying or misrepresenting the Fathers at all for yourself.