So who do you think he's most like?
The most obvious would be Batman. Both Batman and Dresden have a primary city filled with organised crime, that Batman has a tense but occasionally working relationship with in order to focuse on fighting the various chaotic and more insane characters that threaten his city.
As it stands Dresden even has his own Alfred, his own Arkham Asylum and his own Batcave. As well as a League of like minded heroes whom at one point or another were his students to one extent or another.
Of course the primary argument against this is Batmans chief trait as co leader of the justice league which is that he's non superpowered. There's also the fact that Murphy is notably the only member without any superpowers whatsoever, an interest in stopping crime.
Plus there's the fact that Murphy shotgunned Batman when she formed the Dresden League, when Harry died, not disimilar to how the Justice League by Batman is formed in some versions after Supermans death. Though this could also be how Batman was presumed dead and both the Justice League and the Batfamily had to survive without him when Batman is presumed dead.
So is Harry just too OP to be Batman. Which speaking of OP well Superman would definitely come to mind. Particularly how his enemies have to be increasingly OP'd to be a threat to him. Except perhaps Lex Luthor who prefers to use his intellect and way too complex scheme's to ever actually pull off to fight his arch enemy. Not disimilar to Nicodemus, although to be honest why Harry is his arch nemesis when it really should be the KotC is kind of a thematic imbalance, but only slightly as they have been a consistent enemy every five years like clock work.
What both Superman and Batman have in common is that they're guardians of their respective cities that increasingly are left vulnerable by his responsibilities on an international and even interplanetary level. The difference being of course that Batman entrusts his city to his former pupils and allies who are more or less capable of protecting it in his absence, while relying on the Justice League for the international level threats and to some extent the interplanetary ones, while leaving the intergalactic threats to the Green Lanterns.
One other major difference is that so far as to my knowledge neither Batman or Superman have sworn allegiance to nor served any evil cruel torturous or murderous beings in exchange for power. At least not for any lengthy period of time. Although arguably each has done so at one point or another I guess. Batman with Ra's al Ghul in particular. Don't know Supermans lore well enough to pick an example off the top of my head. Did he ever swear allegiance to Zod?
Anyway in the end those are my two votes and why.
(It should be noted that Harry's smart mouth and pop culture references also make Flash and Hal Jordon prime candidates)