"Dying breath" is an example of personification.
Personification is a type of figurative language where something that is not human is described as if it had human qualities. In other words, it is to attribute a non-human entity or concept with human traits. Personification is often used by songwriters, poets, and authors to express a thought more deeply. A metaphor is a comparison between two unlike things, saying one thing is like the other. It creates an image in the reader’s mind.
When you say “dying breath of this love,” it gives the qualities of “life” and “breath” to love. Love is a concept. It is not alive. It does not breathe. However, it is a beautiful phrase that you have here. It implies that love is so real that it is as if it were alive. When you lose someone’s love, it can seem like a death. We often mourn the end of relationships the same way we do people. It makes sense that you could personify love as dying.
Your phrase seems to indicate that love is just barely hanging on to life. This might mean that the speaker really desperately wishes that the person was still in love with him or her. Using the concept of a last dying breath speaks of regrets and lost hope.