Can a repeated random sequence be considered random? Imagine I have a 1000000 bit sequence that is considered random ( ie the output of a TRNG).

Now imagine I 'double' that sequence : I concatenate the same sequence 2 times :  " my 1000000 bits "concat" my same 1000000 bits "

Can we say that the resulting sequence is random ?? ( note that if someone know the construction structure of the concatenated sequence can know the bit 1000001 and following ones as they are the same ones as the beginning ones...)

I have been testing some setups like the one exposed with NIST test suite and seem to show the resulting sequence is random ( anyway im no real expert in NIST test suite so maybe I'm missing some crucial point)
Remember, sequences themselves are not random or not; the process for generating the sequence is random or not.  The question is: is your repeated sequence compatible with the hypothesis that its source is random?

Given a uniform random bit generator, the chance that it will generate any one sequence of $n$ consecutive bits is $2^{-n}$.  The chance that, given that the previous $n$ bits were $x$, a random bit generator will proceed to produce $x$ again as its next $n$ bits is also $2^{-n}$.  So in your case with $n = 10^6$ the chance that a random bit generator is $2^{-10^6}$—a crazy small number.  If we observed such a sequence, then, we would reject the hypothesis that the bit generator is random with near certainty.