When running a Gibbs sampler (for $n=200$ Iterations) with two full conditionals, I  get the output $\mathbf{x} = (x_1^{(n)},x_2^{(n)})_{n =1,...,200}$.

So $\mathbf{x}$ is the realizations of a Gibbs Markov chain, the so called Gibbs sequence. but are $(x_1^{(n)})_{n \in [1,...,200]}, (x_2^{(n)})_{n \in [1,...,100]}$ both realizations of a Markov chain too ?