Is there a reason why every meson and baryon has an integer electric charge?
For example, why is there no meson existing of two up quarks, giving a charge of $\frac43$?
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It is one of the experimental observations that led to the standard model of particle physics. The model has symmetries ( SU(3)xSU(2)xU(1) ) that build up the representations and these allow only for integer multiples charges, i.e. are consistent with observations.
For example, why is there no meson existing of two up quarks, giving a charge of 4/3?
You might say because there is no such a representation in the standard model, but in truth it is because we have never observed such a case. Had we observed it, we would have found a different standard model.