What's the deepest reason why QCD bound states have integer charge? What's the deepest reason why QCD bound states have integer electric charge, i.e. equal to an integer times the electron charge?
Given that the quarks have the fractional electric charges they do, this is a consequence of color confinement. The charges of the quarks are constrained in the context of the standard model by anomaly cancellation, and can be explained by grand unification. The GUT explanation for the charges doesn't care about the bound state spectrum of the QCD sector, so it just seems to be a coincidence that hadrons (which are composite) have integer charge, and that leptons (which are elementary) also have integer charge. 
Now maybe there's some anthropic argument for why such a coincidence is useful (in the case of proton and electron, it gives us atoms as we know them). Or maybe you can argue that GUTs naturally produce fractionally charged particles and strongly coupled sectors, and it's just not much of a coincidence. 
But I remain curious as to whether Seiberg duality, anyons, some UV/IR relationship... could really produce something like the lepton-hadron charge coincidence, for deeper reasons. I suppose one is looking for a theory in which properties of bound states in one sector have a direct and nontrivial relationship to properties of elementary states in another sector. Is there anything like this out there? 
(This question was prompted by muster-mark's many recent questions about fractional charge, and by a remark of Ron Maimon's that the hadron-lepton charge coincidence is a "semi-coincidence", which assured me that I wasn't overlooking some obvious explanation.) 
 A: The simplest answer to your question is a quite old idea, captured best I think by the Rishon model of Haim Harari, Michael Shupe, Nathan Seiberg, and others.
Their answer is the simple and rather obvious one: Hadrons and leptons have identical charge because they are composed out of the same set of more fundamental particles and anti-particles, specifically an uncharged V particle and a one-third charged T particle.
Alas, in terms of mathematical development the Rishon model is more akin to an intriguing speculation than a fully developed and predictive physics model. I do not personally think that any particle-based version of the Rishon model can ever be made to work. My suspicion is that theories like the Rishon model are best viewed as incomplete and distorted images of some far less obvious form of composition, one with components that conserve certain properties but cannot be called particles in any traditional meaning of the word.
Nonetheless, the Rishon model strikes me as orders of magnitude better than some of the more recent trends to explain issues such as electron-proton charge equality by invoking what amounts to anthropic self-selection gone wild. Why? Because Rishon theory at least tries to explain astonishing coincidences. If Newton had given up so easily on looking for deeper roots behind an effect as infinitely precise and in-your-face obvious as electrons and protons have identical charge magnitudes, we'd still be talking about how amazing and lovely it is that Great Angels push the planets around in patterns too lofty and subtle for humans ever to understand.

2012-09-27 Addendum
Here's a point I should make clear for the record, since I came down pretty heavy on the idea that evolving universes could create balanced sets of charges via nothing more than the anthropic principle.
The anthropic observation that the existence of life as we know it seems to require that many fundamental constants to be very tightly constrained and balanced with each other is a simply delightful observation that truly needs explanation. Simple examples include such things as the remarkably long an sharp ridge of stable isotopes that enable complex chemistry, nuclear fusion suitable for stars, and the ability of carbon (with nitrogen and other helpers) to form indefinitely long stable chains. These applications of the anthropic principle are all in effect fine-tuning issues, and I think they are entirely legitimate issues for applying your own personal favorite version of anthropic selection if you are so inclined.
Where I have deep heartburn is with the far more radical versions of the idea that essentially toss all aspects of physics into one big mysterious anthropic pot that then magically burps out whatever it is you need to make life possible. If that is true, why do physics and chemistry constantly throw unexpected structure and marvelous little symmetries in our faces, in even a cursory look? Wouldn't a true, unbiased anthropic cauldron simply toss out a universe that works fine for life, but shows no unnecessary correlations or symmetries between the resulting diverse components of its physics? Such patterns and correlations would after all represent an unnecessary, irrational, and mechanistically inexplicable "extra effort" on the part of the anthropic cauldron, an effort that goes far beyond what is needed simply to enable life. If you own a true anthropic cauldron, Occam's razor says "why bother?" with anything more in the product.
Or stated another way: I have no problem with using anthropic ideas to adjust the ratio between two tightly meshed gears, but I have a lot of trouble with using it to create the gears themselves. Nearly every finding in physics seems to be shouting at us that the bones and tendons of the universe arise from complex permutations and various degrees of breaking of symmetries, with many of details of those symmetries and their permutations being being captured at least partially in that marvelous work called the Standard Model.
So, my real message on this issue is a simple one: Extreme applications of otherwise good ideas tend to be wrong, often rather spectacularly so. Exclusion of extremes is a nicely general principle that applies to a very wide range of phenomena, and I just can't see any good reason why the anthropic principle should get a waiver from it.
A: The universal quantization electric charge definitely isn't a coincidence; it's a necessary result of the fact that the gauge group of QED is the compact group $U(1)$ instead of the noncompact group $\mathbb{R}$, as explained in this extremely short paper by Yang: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.1.2360.
People often act like only the Lie algebra of the gauge group matters, so gauge groups $U(1)$ and $\mathbb{R}$ are equivalent (as are gauge groups $SU(2)$ and $SO(3)$).  But this isn't true: the global topological structure of the gauge group matters as well.  It would be perfectly logically consistent for QED to have gauge group $\mathbb{R}$ and for electric charge to not be quantized, but experimentally this isn't the case.  This ambiguity doesn't come up in nonabelian gauge theory, because in this case the Lie algebra must be semisimple in order for the kinetic energy term to be positive-definite, which implies that the gauge group must be compact.
(In fact, if there were two elementary particles with incommensurate electric charges, implying that the QED gauge group were $\mathbb{R}$ rather than $U(1)$, then magnetic monopoles would be impossible.  Roughly speaking, this is because Dirac's quantization argument giving the allowed magnetic monopole charges requires that electric charge be quantized.)
A: I would say the deepest reason is anomaly cancellation.
If the charge of proton and electron were not the same (even 1 in 1000000!) then the current conservation in the standard model wouldn't be fulfilled due to anomalies.
They say that anomalies have topological roots.
Given the fact that electric charge is quantized in the EM sector (Charge Quantization, Compactness of the Gauge Group, and Flux Quantization), the aforementioned anomaly cancellation technique implies the charge quantization for hadrons that you demanded.
For a more detailed explanation take a look at these pages of Schwartz QFT book:
633 and 634.
A: An experimentalist's view:
I do not see the need to search further for why the three quarks add up to the electron charge than that given by the group structure of the Standard Model. The SM is very successful in organizing  into beautiful symmetries the particle and resonances data gathered the last sixty years or so. There is no experimental reason to assume further layers of compositness defining  a  "deeper" group structure from which the "measured" SU(3)xSU(2)xU(1) should emerge. It will just introduce  a lower level of  unnecessary complexity. 
If what intrigues you is the unit one, after all we can always say the down quark has charge -1, the up quark 2 and the electron -3. The group symmetries are the same  and we will have a generic unit 1 .
