Uniqueness of $W$ and $Z$ bosons What is real reason for considering only ONE unique $W$ and $Z$  bosons, and not of 3 separate flavors, in the Standard Model?
 A: As it happens, there are in fact two $W$-bosons, called the $W^+$ (W-plus boson) and the $W^-$ (W-minus boson). In addition, we have one $Z$-boson, the $Z^0$ (Z-nought-boson).
"Flavour" is a property of quarks, namely it identifies the type of quark that you are considering. In the Standard Model of Elementary Particle Physics (SM) there are six flavour of quark, the Up (u), Down (d), Strange (s), Charm (c), Beauty (b) and Truth (t).
Indeed you are correct in thinking that the Glashow-Salam-Weingberg Model (GSW) which describes the Electro-Weak Interactions is also called Electro-Flavour Dynamics, since it allows the change of quark flavour. Perhaps this is where your confusion stems from?
This process happens through charged weak current interactions. These involve quarks and the charged $W^+$ and $W^-$ bosons.
The reason why there are 'only' three gauge bosons is because of the way that the GSW model is built. Namely, the Gauge Symmetry Group of the GSW model is $SU(2)$. This is the group of unitary $2\!\times\!2$-matrices (see wikipedia) of determinant 1. The group is three dimensional $(2^2\!-\!1\!=\!3)$, and so we get three additional fields.
The story is in fact a little more complicated, and requires the use of the Brout-Englert-Higgs Mechanism (BEH) in order to find the fimiliar $W^+$, $W^-$ and $Z^0$ bosons. Rather, they correspond to a particular choice of gauge, known as Unitary Gauge. In short, this corresponds to the non-zero vacuum-expectation-value of the scalar Higgs boson field being eaten by the SU(2) gauge bosons, such that they aquire a mass term in the Lagrangian of the GSW model, and become what is measured in experiments, such as the LCH at CERN.
The details are a little more technical than this, but hopefully this gives you the idea.
Moreover, hopefully you see why there are only one of each of the $W^+$, $W^-$ and $Z^0$ bosons, since the gauge group $SU(2)$ only gives rise to three gauge fields.
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In addition, note that there is no good reason why fermions come in six different flavour. There could well be sixty of them, as long as they are heavy enough that we have not yet detected them.
As regards "how do we know that there are six quarks?" (that should be, how do we know 'so far'), that famous graph of the ratio of ($e^+ + e^− \rightarrow$ hadrons) to $(e^+ + e^− \rightarrow \mu^+ + \mu^−)$ which shows the steps due to additional quarks comes to mind as a nice way of showing that we have six quarks so far (though it only goes up to b-quark mass). 
It's on page 280 of Griffiths, and I know it's in Martin-Shaw too somewhere. I don't know where it is online though I'm afraid.
To try to address your concern of 
"Why should the boson involved in 
$$ d \rightarrow u + e^- + \bar{\nu}_e  $$
 be same as that involved in 
$$ c \rightarrow s + \mu^- + \bar{\nu}_{\mu}  $$
"
I can only pose the counter-question
"Why should the boson involved in 
$$ d \rightarrow u + e^- + \bar{\nu}_e  $$
 not be same as that involved in 
$$ c \rightarrow s + \mu^- + \bar{\nu}_{\mu}  $$
"
The thing is, there is no need for them to be different!
With the case of the electron and muon, we can measure that they have a different mass, and so they must be different particles. They one goes and adds extra particles into ones zoo. This is how the SM was built up historically. 
No measurement however suggests that there should be an extra $W$  or $Z$  added into the zoo however.
You must recall that the SM is a phenomenonological model. That is, it aims to describe what we measure in experiments, and does a fine job of it too! 
It does not however, aim to explain 'why' any of the things that we see happen. It does not tell us 'why' there are six flavour of quark, 'why' they have such different mass, 'why' a whole host of other things occur. It simply aims to predict what will happen if we have an interaction, given that all of the above properties are indeed the case.
You can see this same sort of difference in, say, Thermodynamics and Statistical Mechanics. Here, thermodynamics gives you some laws, and you can predict what's going to happen in certain experiments, but it does not tell you 'why' 
$$ P V = n K T $$
for example. 
Rather, you need Statistical Mechanics to come along and tell you that, in fact, we have these tiny particles colliding in the gas, and if we take the Hamiltonian, and become aware that pressure is collissions of these tiny particles, and temperature their average energy, it turns out that indeed 
$$ P V = n K T $$
That is, we can calculate it explicitly from first principles. Statistical Mechanics explains why thermodynamics works!
The former case of theory, phenomenonology, is not very satisfying for many people. By the sounds of it, this includes you! 
I certainly find it disapointing to not know 'why' the ratio of electron to muon mass is about $\frac{1}{200}$ say, or 'why' the ratio of the electric charge of an electron to that of a down-quark is $3$, but unfortunately the SM is all we've got! 
There are many people working on various extensions to the SM, hoping to explain some of these seemingly abritrary coincidences. Some make progress, but none are without some flaws.
For example, the SU(5) Grand Unified Theory predicts that the ratio of the electric charge of an electron to than of a down-quark should be $3$. The theory explains where this number comes from.
On the other hand, it predicts that the proton should decay, which we have not yet measured happening. Hence, so far as we know, this theory is not correct.
If you really want to know 'why', I suggest you continue your studies and join us in the good fight to find out!
(P.S., I say 'us' here in a very loose way. It should be noted that I am far from anyone who has contributed something of worth. I don't want that last statement to be read as if I am claiming to have done so!)
A: 
Uniqueness of W,Z bosons

The answer to the title question is that experimentally one SU(2) weak group can represent successfully the observed weak interaction data.This group has the three quantum numbers called "flavors" like an extra charge, to distinguish the three types of leptons and their neutrinos, electron, muon, tau. This SU(2) construct allows for the three gauge bosons, W+ W- and Z0, which are the carriers of the weak force in the same way the photon is the carrier of the electromagnetic force. As carriers they are intermediaries and transfer the flavor from the leptons as necessary to confirm with observations, which is why this SU(2) was chosen. There is no necessity for more as the link you provided allows.

What is real reason for considering only ONE unique W and Z bosons, and not of 3 separate flavors, in the Standard Model?

As explained above, the term "flavor" in physics is uniquely identified as a term to separate the three observed leptons among them, so it is inappropriate, as explained above.  It is the leptons that are sitting in the group representations, not the intermediate bosons which can carry any lepton flavor according to the needs of the interaction.
If you mean more SU(2) groups with other leptons in their representations, the answer is that we have not managed to detect any such, so there is no need for extra Ws and Zs.
A: Well technically if it means that there are different W and Z bosons. Well there are two W bosons. The W+ and W- bosons. Also on top of this these bosons normally can have different masses. However usually different masses are not really considered for why there are many types of W+, W-,and Z bosons. The Z boson normally has a mass of around 90 MEV. However this can change depending on the interaction, and whether it is a virtual or real particle. If they categorized it as different weak bosons then there would be so many particles to account for. So that is why it is just considered as just two W bosons and Z boson. As a matter of fact sometimes the standard model just treats W+ and W- bosons as the same boson on the simplified standard model. That is because the W+ and W- are antiparticles. Though it is not considered they are antiparticles.
