The receipt of a gift concludes the act/transaction of giving. There is no give-backs, except when a gift is not just a gift, and that is when something is expected in return. In such a case, the donative intent is missing, the giver does not just intend on leaving the gift with the receiver.
The engagement ring is a conditional gift. It is given upon the condition that a marriage will follow. If there is no marriage, the condition is not satisfied, and the ring should return to the donor.
As opposed to the birthday gift scenario, the giving of the engagement ring is not the start of the act of giving. The engagement process starts with a question, will you marry me? Only upon an affirmative answer, yes, do you give an engagement ring.