"My True Love Hath My Heart" by Philip Sidney uses the literary device of metonymy. Metonymy is when someone writes about something, but does not use its name to describe it—instead he/she uses something associated with it.
For example, when we say we could eat the whole plate (of cake), we are not referring to eating a plate, but the food on it.
For this poem, the metaphor I find is that the heart represents love. The metaphor is that the heart is said it is "given" like a tangible item, when it really refers to sharing an emotion, something intangible.
Each lover has shared his/her heart (love) with the other, and the other, in turn, has done the same. "He loves my heart for once it was his own." So the two have shared their love for one another. In doing so, the love has altered, become something new in combining the two separate loves into one.
The exchange of love, represented by the heart, is shared between these two, but it's hard to tell where one person's heart stops and the other person's heart begins. They become like two hearts in the same chest.
The best way for me to make sense of this is to suggest that when one gives his/her heart to the other, they become inseparable: not physically, but metaphysically. The essence of one joins with the essence of the other and they cannot be distinguished from each other any longer.
When Elizabethan's married they believed that at that moment, they were inseparable forever. This is why in Hamlet, by William Shakespeare, Hamlet's mother is committing incest by marrying her brother-in-law: for within Gertrude there still resides a part of her dead husband (Old Hamlet)...therefore, Old Hamlet is sleeping with his brother (Claudius) when Gertrude sleeps with Claudius.
As Sidney lived at the same time as Shakespeare—during the Elizabethan period—I believe he would also have been influenced by Elizabethan perceptions: it makes sense to find this kind of "imagery" is used in Sidney's poem.
The idea of oneness guarantees that each person will do all he/she can to make the other happy, for each has a part of the other's heart, the other's love. And hurting the other would, in fact, hurt him-/herself.
Like a house of mirrors—it is a little hard to follow. Hope this helps.