Why are all quarks and leptons of this universe the same? We know the composition of stars by spectroscopic analysis. The EM waves generated by them are blue- or redshifted. We could have said, "Look, the wavelength is slightly different so it may be constituted of quarks and leptons which are slightly different than ours."
How do we ascertain that it is made up of the same particles?
 A: Fairy Physics
It is entirely possible to construct a theory of the universe which states: "All effects are caused by fairies.  Each effect has its own fairy, and every fairy is unique.  When two fairies produce the same outcome, that is just a happy coincidence."  Unfortunately, it is basically impossible to disprove this theory.  Also, this theory lacks explanatory power.  If I ask the question: "Where will Venus be in 3,000 years?" Fairy Physics can only answer: "Wherever the Venus fairies decide it will be!"
And this is the real problem with Fairy Physics.  The problem is not that it is "wrong", because in some sense, it can't be wrong.  The problem is that it is useless.  And it is useless because it is, in a way, infinitely powerful.  It is a theory which allows anything to happen, because it's answer is always: "A fairy caused that."
And thus, we see that a useful theory is one in which we can separate "explainable" events from "miraculous" ones.  There are no "fairy miracles."  But if we witnessed Jupiter teleport to the other side of the sun that would pretty fairly violate the Standard Model of physics.  A theory actually becomes more powerful the more constraints it imposes on how the universe can evolve.  That's because more constraints means a greater ability for us as human beings to predict the future.
Physics has proceeded by imposing ever more detailed limitations on how the physical world is predicted to behave.  A universe in which quarks and leptons differ across space and/or time has fewer constraints than one in which quarks and leptons are identical.  And thus, a theory which describes such a universe is weaker than one which forbids it, because it allows more behavior.
Standard Model
The nice property of a constraint is that it gives you a way to invalidate a theory.  The more specific and precise a prediction, the more ways it can fail.  And if a prediction succeeds, you thus have more confidence in the prediction under tighter constraints.  Fairy Physics is "true" because it cannot be falsified.  But such theories are, as we have established, utterly useless.  We want a theory with the tightest possible constraints we can impose, because such a theory offers the sharpest predictions and the most opportunities for falsification.  If observations are then compatible with the resulting theory, we have much greater confidence in it.
The Standard Model is widely accepted because it offers the strongest possible predictions we know how to make, and our observation of the universe does not provide any strong counter-examples to falsify it.  Standard Model-- with bespoke quarks and leptons elsewhere in the universe is a weaker theory, and also unnecessary.  So why downgrade to a Ford Fiesta when you can drive around in a Lamborghini?
To give a more explicit example, consider MOND: Modified Newtonian Dynamics.  This is a possible explanation for Dark Matter which causes gravity to behave differently on large distance scales.  But even this theory avoids letting gravity simply vary arbitrarily across space, because such a relaxation would be giving up too many constraints.  An isotropic, homogeneous universe w.r.t. the laws of physics offers the strongest constraints for a physical theory, which is why almost no modern theory will give it up.  Doing so cripples the theory to a level that few find acceptable.
A: Actually 2 questions:

*

*The title: why are leptons and quarks all the same? We don't know, but we have a good model for them and it works pretty well. In this model, particles emerge as a solutions to a certain equations. We more or less can imagine what would happen if we change the parameters of these equations (say, fine structure constant) and the result is nowhere to be seen (distant past on Earth, distant galaxies, etc...).

See here for an early attempt to explain why electrons are all the same.


*Can blue/red shift be explained by alternative means? Sure - and everyone is free to attempt. If someone presents a better theory (better as in explaining more of the observations and the experimental results) it will enter the textbooks right away.

The problem is, blue/red shift is very good at explaining speed and gravitational effects all the way down to our best measurement abilities. That's why we use it and when applied to distant stars, the particles look just like our own.
In order for a new theory to evolve, one have to measure something with better accuracy and prove it deviates from the established theory.
A: The reason has to do with quantum field theory (QFT). In QFT, particles are manifestation of excitation of the fields. These excitations are quantized and are universal in nature. Since the laws of physics are true everywhere in the universe (except for in blackholes probably. We don't have a clear idea what goes on inside it yet), these excitations are identical as well. The consequence of this is all particles of a particular species, are identical in every way. If you want to know more on this subject, I request you to go through some QFT books. It might be a tedious thing to do but this is the only way to get some honest idea about your nice question.
A: I want to expand on the comment I made to gandalf61 's answer.
Spectroscopic binary systems are usually detected due to the movement of the emission and absorption lines in the observed spectrum, caused by the Doppler effect as the stars move in their orbit.
This shows experimentally that even though the two stars are far away and moving with their center of mass velocity, the spectra are affected by the velocity vector, showing a periodic difference in the doppler shift.
So even if we know nothing else, we can experiment in the laboratory to see if different values would need to be attributed to quarks and leptons for different velocities. Needless to say that the standard model axiomatic attributes as given in the table fit the data very well within measurement errors. Thus the assumption holds that velocity does not change the attributes of quarks and leptons.
So the observation from the binary systems of spectral line changes due to velocity vector changes is expanded to velocities of the  luminous bodies in our observable universe.
A: We observe that in the neighbourhood of our solar system, physical laws appear to be the same everywhere. We then observe that this appears to be true for the whole galaxy.
However when we look further afield, we notice not only the redshift which you mention but also that it appears to increase more or less proportionally with distance.
For the present purpose this cosmological redshift offers two principal explanations. One is that, assuming a constant size to the Universe, the laws of physics change proportionately to their distance from us. This places us at the epicentre of spherical shells of change, an extraordinarily unlikely event. The explanation involves the argument that we are only here because it is only at this epicentre that life is possible; this is known as anthropic reasoning and widely condemned as pseudoscience.
So what about that other explanation? This is, that the laws of physics are in fact the same everywhere and that the Universe is everywhere expanding at a constant rate. Even though the expansion makes no sense and even seems to be slowly speeding up, we choose to waffle vaguely about "inflation" with no idea what that really entails. This is still far more acceptable to the scientific community.
Personally I think that apologists for the current model are a little over-confident. One may ask, why then are we here on this planet and not another one? The answer if of course that only this planet supports complex life, but on what basis is that not anthropic reasoning, while a centre of physical law variations is? Then again, this inflation thing. The theory has been around half a century yet remains as vague as ever, with no sign of the mythical "inflaton" particle nor any other faster-than-light expansion of spacetime (nor even a satisfactory explanation of how that last is not an oxymoron; brushing it off with a glib "there is nothing to prevent it" risks placing it conceptually in the realm of angels and the afterlife). Moreover, there have been many suggestions that the laws of physics, such as the fine structure constant, might vary with time. And, due to the finite speed of light, looking outwards is also looking back in time. So who is to say that these shells of spacetime really do embody constant laws? The anthropic problem is resolved because we are judging the past by today's laws, with each shell representing a step back into our past; their centre is thus relative to the observer and not an absolute epicentre after all. Any such variation is expected to be far smaller than the redshift observed, but who can say for sure? I am not saying the standard model is wrong in this respect, but I am suggesting that we don't know enough to be entirely sure.
A: We know that the spectral lines in the spectrum of a binary star shift one way and then the other and this is correlated with its position in its orbit around its companion. Clearly, the constituents of the star do not change with each orbit so the shifts in spectral lines must be due to the Doppler shift. Occam’s razor then suggests that we apply the same explanation to all spectral line shifts instead of requiring different types of fundamental particles.
A: Other answers have correctly given some of the reasons. I am merely going to add that ultimately this is a question that bears on a very large number of different observations, on the concept of constructing a model, and on Occam's razor.
The observations are the accumulated data from all of astronomy, combined with knowledge of matters here on Earth, such as the spectra of the chemical elements and things like that.
The big picture is that we construct a model. That is, we make a hypothesis about what the universe is like, and deduce what would be observed if the hypothesis is right, and what kind of observations might show the hypothesis to be wrong if it is wrong. One such hypothesis is that the fields studied in quantum field theory extend everywhere, with the result that leptons elsewhere have all the same properties as the ones near to us. This then underpins a model. Once combined with general relativity, the model is consistent with observations and is as simple as we know how to make it such that the consistency holds. If anyone wishes to doubt the model, they are free to do so, and invited to propose another model. If some other model matches observations and is deemed simpler than the one currently having wide acceptance, then the other model will, either quickly or gradually, gain acceptance.
What this means is that if someone wishes to propose that leptons elsewhere are different to the ones near to us, then they are invited to make that proposal more concrete by showing how it is part of a consistent and elegant model of the relevant quantum fields etc. It is VERY hard to do that. Theoretical physicists do try to explore models of this kind, but the proposal runs into all sorts of difficulties and it seems that it can't be made to work. So this amounts to evidence, or a suggestion (not a proof, not a demonstration) that the standard model is right (I mean the one in which leptons elsewhere are like the ones near to us, and the universe is expanding), or at least it is right in this respect: that quantum fields are the same everywhere, or at least in most of the observable universe.
A: There is already an excellent answer given by @Lawnmover Man. I would just add a comment from a point of view of probability theory and machine learning.
If you have some set of observed phenomena and a different models which explain these phenomena, in addition to the fact, how well our model is consistent with the observations, one should also pose a question - what would happen, and how would our model react, if it sees something new, different from the previous data. On the available by far data, the model, which gives a possibility for matter in other galaxies to differ from those in Milky-way, and at the same time neglect redshift-, blueshift- phenomenon may fit the observations rather well, but at the same time it would be less beatiful and convenient, from the point of view of physicist, because it would require estimating the parameters for each galaxy or space region. And eventually, for the case of extremely large redshift-, this may be not even physically constistent with the other considerations.
From the probabilistic point of view, we may assume, that the physical constants come from a certain distribution over parameters, and the values, we have in the nature, arose by a coincedence by toss of a coin. If we have a large number of parameters, the specific configuration of parameters and constants becomes less and less probable.
A: Because if weren't, there would be domain walls between the volumes of different particle families.
And the domain walls would radiate because of the interaction of different particles.
