What is the definition of a qubit and a copy/clone of a qubit? A qubit with state $|\psi \rangle  =\alpha|0\rangle + \beta|1\rangle$ is defined as : if we have infinite copies of $|\psi \rangle$ and measure them all in the basis $\{|0\rangle,|1\rangle\}$ then $|\alpha|^2$ percent  of them would measure $|0\rangle$ after measurement and $|\beta|^2$ percent of them would measure $|1\rangle$. So definition of a qubit with a particular known state depends on the definition of copies of a qubit ( clone/copy ).
We define two qubits A and B being copy/clone of each other if, same percent cent of qubits give same measurement results if we take infinite copies of qubit A and qubit B separately and measure them in $\{|0\rangle,|1\rangle\}$ basis. So definition of being a clone/copy is dependent on its own definition. Thus even the definition of a qubit with a specific known state appears recursive to me. What am I missing ?
 A: A qubit is simply not defined the way you define it, neither are clones. This is mostly, because you describe a state by referring to a state. It's not the problem that you are actually referring to copies of the state, you are referring to copies of the state, which is yet to be defined.
Let's go to what I have learned as the Ludwig school of thought about states and experiments, which is an operational definition (and therefore probably closest to the kind of definition you are searching). 
We have to start out with a physical experiment. How can I describe this? Well, in an experiment, you have a stage where you prepare your system. After the system is prepared (e.g. creating a beam of polarized light with a laser beam and a polarization filter for instance), you measure it. If you repeat your measurement, you'll get a probability distribution over your measurement outcomes. 
Now the key thing to realize is that the preparation procedure describes the state of a system. In other words, a state is an abstract description of how to actually create e.g. a particle with certain prefixed properties. A state is something like a class in object oriented programming and it must not(!) be confused with an instance of that class. In your example, a qubit might be described by the procedure of how to prepare a photon in a superposition state of up and down-polarized light ($|0\rangle,1\rangle$) according to some parameters ($\alpha,\beta$). The single photon is not(!) a state, it is only an instance of the state. 
In everyday language, we often confuse the concept of a state and its instances, because it rarely matters, if we know how to interpret the probability distribution produced by a state (which, if we see the state as a preparation procedure, becomes the asymptotic empirical distribution of the instances of this state), but to me it seems crucial to highlight the problem with your definition of states.
Note that until now I have not even mentioned measurements - that is, because they are not important for the definition of states, they form something like its dual part. Now, if you want to compare two states, because, say, you have two preparation procedures and want to determine whether the abstract state behaves the same way, then you need to consider measurements and you would say that two states are equivalent if their empirical distributions according to an informationally complete measurement are the same in the asymptotics of large numbers of instances of the state. [As an aside, note that your measurement procedure above is NOT informationally complete: You do not retrieve the phase information, you can only compare $|\alpha|^2$ and $|\beta|^2$. To be able to say that two states are equivalent, you will have to know the relative phase between $\alpha$ and $\beta$]. It seems to me that what I call "equivalent states" are what you think of as "copied states". 
But then, what are cloned states? In fact, we would not define a "cloned state", but a "cloning machine". A cloning machine takes an instance of the state as an input and outputs the same instance and an instance of another state that is defined by this procedure. If the corresponding states are the same, then we consider the machine a "cloning machine". In other words: We have two states, state A which we start out with and another one, state B, with the preparation procedure: "prepare an instance of A, send it to the cloning machine and take the outcome". Now, if these two states are equivalent, then we would say that state B is a clone of state A and the machine was a cloning machine. If we could find a cloning machine that produces a clone for whatever input state I define, then this would be a universal cloning machine and as you very well know, this is forbidden by quantum mechanics.

Now, I said above that this is what I came to know as the "Ludwig school" of thought (an operational approach to quantum mechanics, which was mostly developed by the German physicist Günther Ludwig and his students). 
A different definition would be, if you consider the statistical interpretation of quantum mechanics. Then you'd never have a single instance of a state, your "state" would rather be a statistical ensemble of many such instances - at least that's what I think you do.
Another approach would be to just see the "state" as an abstract description of what I called an "instance of a state". It is just not defined operationally. You'd interpret Born's rule as saying: Take a state, measure it. If you perform this experiment more often, then the empirical distribution of the outcomes would asymptotically be given by Born's rule. It does not(!) say: Take copies of the state. It says: Do the experiment all over. If you think about it, then you'll see that this is pretty close to what I described above, because this actually means "prepare the state and measure it".
