“…we need another soul to cling to.”
Victor Frankenstein’s monster “grows up” without any kind human contact. His entire socialization process happens via observation. His observation of family life and love gives him a desire for that love and companionship for himself.
This drives the creature to asks Frankenstein that he “must create a female for me, with whim I can live in the interchange of those sympathies necessary for my being.” He claims that his “monstrosity” and maliciousness comes from his misery and loneliness.
The creation is shunned by all of humanity and feared by those who see it. Even the creator has an intense desire to destroy his own creation and it is not because of his monstrosity that he is hated but his monstrosity emerges because he is hated and feared. This creates a vicious cycle.
The logical solution, then, would be for him to retreat from human society and live out his life in solace and isolation. This might have been possible initially but his experience in observing the lives of normal humans has awakened in him the desire for companionship and possibly, love.
This drives him to request his creator for a companion. It is interesting that he should ask for a female companion. This could be because he has certain sexual urges or because he observed the love between Safie and Felix and wanted something similar. It is not very clear as to whether the monster has received any knowledge about where babies come from so his request may simply be innocent or it could be a part of a plot to procreate and destroy all of humanity as part of his revenge. The creature has shown himself to be very patient.
The motivation behind the request for a companion makes sense but it definitely does not make complete sense. Shelley has introduced quite a lot of ambiguity there, perhaps intentionally and that decision leaves a lot of space for interpretation.