There are many versions of Joker in Batman lore, but most have zero interest in Bruce Wayne. Why doesn't the villain care about Batman's identity?
Why is it that so many versions of The Joker are either indifferent to knowing the identity of Batman, or actively try to avoid learning the truth? Throughout the Batman stories of comics, screen and video games, many different Jokers exist, ranging from dark failed comedians to flamboyant tattooed gangsters. In many ways, The Joker is a character defined by his inconsistency, but one common trait many iterations share is their general disinterest in "Bruce Wayne." While a few Jokers have sought to learn Batman's identity, most notably in Batman Beyond where the villain tortured Tim Drake, Joker traditionally prefers not to see under the mask. This reluctance is demonstrated in The Dark Knight, when Heath Ledger's Joker puts out a hit on Coleman Reese, the Wayne Enterprises employee (there's a clue right there) who claims to know Batman's identity. Joker believes Batman's public outing will ruin the fun he's having with the Caped Crusader, and this attitude broadly follows the DC comic books.
There are several instances of Joker confirming or hinting that he knows Batman's identity in the source material, for example when Bruce Wayne hands back one of Joker's calling cards in Death of the Family, annoying the imprisoned villain. Arguably, Joker's lack of interest in Batman's identity doesn't make sense. Think of all the extra fun and games that would come from torturing Bruce Wayne as well as Batman. Thinking practically (if Joker is capable of doing so), the villain may not want Batman's identity revealed because it could mean the end of the superhero's career, and Joker losing his best enemy. But since Joker seems more personally aggrieved to find out Batman's real face, and not as intent on preventing others from discovering it, there must be more to his reasoning.
The Dark Knight's Reese story hints at why The Joker doesn't want to know who Batman really is, but his true feelings on the subject go even deeper and concern the opposing philosophies of DC's biggest rivalry. Joker is one of the few Batman villains who doesn't have a proper civilian identity, and this is a deliberate choice. The Joker is The Joker 24/7 because whoever he was before made that wholesale transformation, but Bruce Wayne did exactly the same thing when deciding to become Batman, even if he did maintain a dual identity. Just as Joker left behind his former persona to evolve into a maniacal clown, Bruce did the same to become Gotham's protector and The Joker is undoubtedly more aware of this shared duality than Batman is. To Joker, Batman is real and "Bruce" is the masked alter ego.
This dichotomy is neatly summarized by Grant Morrison in Batman #663, when The Joker says to his nemesis in "The Clown At Midnight" "Why be a disfigured outcast when I can be a notorious Crime God? Why be an orphaned boy when you can be a superhero?" In this story, Joker's lack of interest in Bruce Wayne is the same as his disinterest in his own life before becoming the Joker. Morrison hits the hail on the head yet again in Arkham Asylum: A Serious House on Serious Earth, when Joker says of a cowl-wearing Batman, "that IS his real face!"
A separate strand to Joker's Bruce Wayne aversion (once again alluded to in The Dark Knight) relates to his overarching mission and obsession with Batman. Where early versions of the character depicted another criminal being constantly apprehended by a vigilante in tights, The Joker and Batman's relationship has evolved into a deeply personal one in which both characters represent far larger ideals. As before, Joker shows much more awareness of this dynamic than Batman does. The villain represents chaos, disorder, and the inherent fragility of societal norms, whereas Batman symbolizes the opposing qualities of good, justice and selflessness. As a beacon of these traits, Joker and Batman are natural enemies, but Bruce Wayne, from a public perspective at least, represents wealth, lavishness, industry - things that Joker has little interest in. The Joker could attack these ideals by targeting any one of Gotham's elite, but there's only one Batman, which is why Bruce Wayne will always be an unnecessary sideshow to the iconic Batman villain.
