It is a personification. A personification is when traits that are usually given to human beings are applied to objects or ideas. In this example, love is described as if it had breath, like a person does, even though, of course, love does not have breath at all. This helps tie the reader more closely to the object of comparison, since as humans, we tend to see things better in human terms.

Metaphors are a comparison in which there is no comparative language used. The writer directly refers to something as if it were the object of comparison. For example, if you wanted to use a metaphor to compare a tree to a castle, you would say, "the tree is a castle," rather than, "the tree is like a castle." Terms such as "like," and "as" are not part of metaphors. They are used in another form of comparison, known as similes.

"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.