Yes! The repetition of the word "mine" in "Mine, mine fair" is significant. In repeating the word, the persona reflects backward (to "that moment she was mine") and forward (to "mine fair,/ perfectly pure and good"). The backward reflection suggests his desire for ownership while the forward "mine" suggests a desire to possess a state of being (her fair, good, pure). By repeating the word, he tells us that it's not all about having her, but that it's also about preserving her perfect state--that moment of pure love for him.
That's the crux of his problem--he knows she loves him; it's just not enough.
He has told us already that although she loves him, she is not sufficiently committed to him. She wavers. He says she is too weak to follow her passion and to give herself to him forever (21-25). When he says she is "mine, mine fair", he is saying that he possesses her heart in its most perfect state. Were he to let her live, he would have her body sometimes, and her heart, sometimes, but not always.
Here's where the second aspect of the repetition of the word mine comes in. The repetition also serves to make him appear somewhat child-like, greedy and immature, someone who won't share, who wants the perfect doll still in its box. He wants something that is his alone, something that no one else can have, something that she , were she to live, could never be.
The poem is a dramatic monologue, and dramatic monologues always reveal as much about their speaker as about their subject: they show, rather than tell. It is the case here, too. Browning's repetition of 'mine' as the speaker relives the moment when Porphyria declares her love for him (the poem's in the past tense, remember), creates the sense of his speaker's awed wonder and also his possessiveness. This potent combination, added to his sense of social inferiority to Porphyria herself, makes it a short step, given the speaker's disturbed state, to preserving that moment forever by killing her. At the end of the poem, when the speaker declares that 'God has not said a word!', he seems triumphant, and that sense of arrogant power is anticipated by the repetition of 'mine' much earlier in the poem
It is in lines 31 to 36 that the speaker suddenly uses a series of first-person pronouns, such as "mine, mine". This is when he realizes that Porphyria belongs to him completely. He has doubted this up to this point because Porphyria is from the upper class, and her standing in the society has concerned him as to whether she loves him. She is superior to him socially and shows her power over him. When he realizes that she "worships" him, he wants to keep that forever, so he kills her. By feeling she now belongs to him, he kills her. He is then able to become the dominant one in the relationship. He is now in control. "Mine, mine" shows his moment of truth where he realizes he must kill her in order to keep her forever.
Remember, the speaker of the poem is insane. We have only his point of view on what happened. We don't get an objective viewpoint of what Porphyria is like or what her feelings for him really are. It is in his madness that he feels he must kill her to keep her as his love forever.