Does saying "May God forgive you" to someone who caused anger, without internal forgiveness and a desire to obtain one's right, relinquish that right on the Day of Resurrection, and what is the meaning and function of this phrase in religion?

The phrase "May Allah forgive you" is a supplication for the addressed person to be forgiven. The speaker should have already forgiven their own right for the supplication's intent to be fulfilled, because Allah does not forgive the rights of His servants until they forgive first. Sometimes, this phrase is said without its original intention. It is better for the wronged person to forgive and overlook for the sake of Allah, and they will have a great reward, as Allah Almighty says: "And to forgo it is nearer to righteousness" and "and who restrain anger and who pardon the people - and Allah loves the doers of good" and "So whoever forgives and makes reconciliation - his reward is due from Allah."