What happens to the data distribution and results if we calculate z-score of a z-scored data? The data that I am using is already z-scored and batch normalized. I accidentally calculated the z-score again and then performed further analysis and calculated results. Does it make sense to take z-score of a z-scored data distribution? Does it affect the distribution and the results? Thank you

Pretty much nothing happens. The mean of your z-score data is $0$, and the standard deviation is $1$. Thus, when you run your points through the formula, you subtract $0$ and then divide by $1$.
$$
z=\dfrac{x-\bar x}{s} 
$$
Because you're likely doing this on a computer, it is possible that your $\bar x = 0$ and $s=1$ are stored as something like $0.0000000710008$ and $1.00000004000502203$. I would ignore that in every situation that comes to mind. Because you did something unusual, however, my recommendation is to redo your work with only the one z-score calculated, knowing that almost nothing is going to change. (You are setting yourself up for criticism that you can avoid.)
A simulation in R shows all $1000$ second z-scores to be within about $10^{-16} = 0.0000000000000001$ of the first z-scores. Since I simulated an exponential distribution, this fact does not depend on the data being normal.
set.seed(2021)
N <- 1000
z_score <- function(x){
  return((x - mean(x))/sd(x))
}
x <- rexp(N)
z <- z_score(x) # Calculate the z-scores
z2 <- z_score(z) # Calculate the z-scores of the z-scores.
summary(z - z2)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
0.000e+00 9.368e-17 1.110e-16 9.113e-17 1.110e-16 1.110e-16 

