Your statement is correct in terms of the inexact nature of the monster. Chapter IV of Mary Shelley's novel Frankenstein explains in detail the way in which a then-obsessed Victor Frankenstein collected what he calls his "working materials", in order to create the body that he intended to infuse with life.
I collected bones from charnel-houses; and disturbed, with profane fingers, the tremendous secrets of the human frame.
With this information, it is true that the argument that nature built the monster's identity would be nullified since there is no uniformity nor predictability that can be gathered from the genetic origin of each of the creature's body parts.
However, what if your argument by-passes genetics and moves into the phenotypical features of the creature? Think about the unique nature of the monster; it does not belong to any known species. It is still "human" (based on its genotype) but it cannot be classified under any human species category. And yet, even though it is an abnormal creature, it still is a social creature with a clear need of acceptance, validation, and companionship.
The information above can help you create a new argument, still in favor of nature: that the monster's abnormal looks, its horrid features, and the carelessness of the manner in which it was put together caused the primary trauma of its life and are the reason behind his identity.
In other words, that the first shock in the creature's life was not the way that he was treated, but the realization that his appearance was unnatural, inhuman, and revolting. Any normal individual can tolerate the injustice of being mistreated if there is a backbone of support to which one can return to reinforce self-assurance, awareness, and dignity. Yet, the creature cannot do that. When he realizes "what" he is, and not "who" he is, all of his internal thoughts and feelings become hardened with the question of identity.
As a result, he is forced to adopt an identity based on what he represents in the eyes of the world; he cannot be or act like a normal man, for he is not welcome in normal society. He cannot opt to act like an animal, because he is not that, either. Hence, his only choice is to identify himself as what he appears to be: a monster.
To support your argument, refer to Chapter XV of Frankenstein here on Enotes. This is the part of the novel where the creature finds the manuscripts of his creation, and finally understands his true nature; notice that, although the creature had been suffering from the treatment of others, he still does not really realize what he is until this point. This is the moment where his identity as a monster is finalized; it is here when he finds out that he really was meant to be a sort of monster by his own creator. It is all explained in the following quite.
‘Hateful day when I received life!’ I exclaimed in agony. ‘Accursed creator! Why did you form a monster so hideous that even you turned from me in disgust? God, in pity, made man beautiful and alluring, after his own image; but my form is a filthy type of yours, more horrid even from the very resemblance. Satan had his companions, fellow-devils, to admire and encourage him; but I am solitary and abhorred.’