Is "dying breath of this love" a metaphor or personification?

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.