The following is from page 139 of the book Apostol's Analytic Number Theory calculating Dirichlet characters mod 5 and 7:
My question is that can any of the rows inside the red squares be exchanged with another one, with $\chi_i$'s fixed? (for example what are written for $\chi_3$ can be written for $\chi_4$ and simultaneously vice versa.)
I think you are really asking about whether Apostol is using some sort of cannonical labeling for the Dirichlet characters, so that the label $\chi_2$ has some semantic meaning in itself.
But in Apostol's book, there is nothing special about the subscripts. So the character $\chi_2$ could just as easily have been called $\chi_3$, and vice versa. The principal character $\chi_1$ is special. And in these tables it's clear that $\chi_2$ is special in the sense that it's "simpler" than the others. But that's it.