This epigraph is best dealt with by being broken down into two parts, but there are vague links between Hardy and Shakespeare's works tied together in this particular quote. (Also, it relates to the overarching themes and events in Tess of the d’Urbervilles, so this answer will link it to such rather than specific quotations from the texts to avoid this answer becoming overburdened with perhaps irrelevant degrees of detail).
Firstly, “Poor wounded name!” corresponds to Tess’ damaged reputation on account of her extramarital pregnancy as well as her unavoidably being a murderer - her name becomes synonymous with her social outcast. The metonym of focussing on a name and not physical person was highlighted by Rıza Öztürk in The Origin of Hardy’s Tragic Vision “might be associated with Shakespeare’s emphasis on the character” (p. 66); since characters in novels are no more than words, Tess only has her name and since it is tarnished, her whole being likewise is. This “wounded” status is not from the self but from external forces, as Tess has been used and discarded by both Alec and Angel and is also subjected to the scorn of Victorian society. This mirrors an arc in The Two Gentlemen of Verona: Julia, whom Proteus loves, receives a letter from Proteus which she tears up over a disagreement, an action she deeply regrets later on in the scene, causing her to utter the monologue containing the lines of the epigraph. Julia’s behaviour following her tearing up the letter suggests that she feels remorse appropriate to having physically injured Proteus through the action (as hinted at with its use in Tess of the d’Urbervilles):
O hateful hands, to tear such loving words!
Injurious wasps, to feed on such sweet honey
And kill the bees that yield it with your stings!
I’ll kiss each several paper for amends.
Act 1, Scene 2, lines 110–113.
The injuries to Tess are on a grander scale than the largely domestic squabbles of Shakespeare’s play. Regardless of whether she conceived by a consensual act or not, Tess was still subjected to the social disdain that was directed to her after she gave birth to her and Alec’s child outside of wedlock. Likewise, Tess is executed as a criminal when, in desperation, she murders her second husband Alec, after her first husband Angel returns from what she thought was the dead (actually only Brazil) since it had appeared that he was not returning after effectively abandoning Tess, upon discovery of her earlier affair with Alec.
The next phrase of the epigraph leads on from this: “My bosom as a bed”. The “My”, positioned before the novel’s opening cements it as Hardy’s voice, overriding any voices in the narrative, since it is placed before any input from the characters can arrive on the scene. For all of the instances where Tess’ honour comes into question, Hardy seeps his voice into the narrative as the background narrator of the events. This is established from the foremost beginning of the novel in the subtitle: “A Pure Woman Faithfully Presented, by Thomas Hardy”. Hardy places his name seamlessly into the subtitle here and boldly goes against the grain of society’s judgement and emphasises his opinion (reference) of Tess as a blameless victim, acting according to human error and natural behaviour, not as a perpetrator of unnatural crimes. As a result, Hardy singles himself out as Tess’ sole protector and advocate in what is a preemption of society’s criticism that would flood the novel at the time of publishing, certainly in part regarding the depiction of Tess’ social downfall, but specifically Hardy’s personal and heartfelt sympathy to it. The clarity he uses in making sure that Tess is regarded as a “Pure Woman”, despite being the mother of an illegitimate child as well as a murderer and adulterer, is what would have been seen as a very adverse viewpoint. Hardy verbally defended Tess’ purity in an 1892 interview, attributing her outwardly-appearing downfalls as whims of fate and buffeted by external circumstances:
I still maintain that her innate purity remained intact to the very last; though I frankly own that a certain outward purity left her on her last fall. I regarded her then as being in the hands of circumstances, not morally responsible, a mere corpse drifting with the current to her end.
As cited and further elaborated on by Merryn Williams in Thomas Hardy and Rural England, p. 91.
However, Hardy unashamedly defends this by further claiming it to be “Faithfully Presented” aiming to dispel any distortions of biased perspective, by suggesting that the following narrative is an autobiographical representation of the unfortunate life of Tess, based in reality and not a fictional creation, which subsequently labels the judgement of Tess as a “Pure Woman” by Hardy as definitive, without any shadow of doubt to its truth or authenticity. On a broader level, Tess is Hardy’s creation, which could tenuously link to the imagery of the epigraph as she is consequently inspired from the “bosom” of his imagination, a protective place of authorial defence, to which she can return to “lodge” following critical exposure to readers of the novel. However, this is of course a metaphorical interpretation and Shakespeare employs this part in a very literal sense in The Two Gentlemen of Verona, where Julia places the torn pieces of the love letter into the fabric of her bodice.