Does existence of various kinds of quarks like up, down, strange, charm, top, bottom violate the quantisation of charge or just redefine it as up quark have charge +2/3 and have -1/3. Or do things get even complex for unified theories like the proposed string theory?
That is a good question but I think you might be a bit confused. The quark charges are quantised as they are fractional values of the electron charges, so when you refer to 2/3 and -1/3 these mean 2/3 of the electron charge and -1/3 of the electron charge respectively.
As such, a Hydrogen atom with a proton in the nucleus and an electron in the shell, is electrically neutral:
Proton = 2 up and 1 down which means 2/3+2/3-1/3=1, or exactly one (positive) quantity of the electron charge. As the electron has -1 quantity of its charge, the overall electrical charge balance is vanishing and the Hydrogen atom is neutral.
The rest of your question points out to a bigger question, why are the quark charges fractions of the electron charge?
This question is still unanswered, but there are many possible explanations. The most common one is the one provided by Grand Unification Theories , which for most constructions exactly predict these fractions. It is interesting to point out that Grand Unification Theories are naturally present in general frameworks such as String Theory.
Other thing to notice is that if the the quarks were not to have these charge assignments, then the Standard Model of Particle Physics would have the so called anomalies  and the theory would be inconsistent. Notice, though, that this is not an explanation, but rather a consistency check.
The existence of quarks is not seriously in dispute at this point AFAICT. If you want to make something meaningful out of quarks and only quarks having fraction-of-$e$ charges, I think you pretty much have to postulate that electrons are composite.
For instance, the rishon model proposes that all the "fundamental particles" of the Standard Model are composites of still-more-fundamental "T" and "V" particles and their antiparticles. T has a charge of $e/3$, V has a charge of zero.
This model is fairly old as these things go (1979) and may well have been falsified by things like the current bound on the electron dipole moment, the existence of the Higgs, the continued lack of evidence for proton decay, etc. — I'm only mentioning it as an example.