I dont think this is a correct understanding.
The best way to see it is to count the number of free parameters. Here e(j) has 10 elements, which indicates there is 10 “draws” or “samples” from 1 prior distribution. Each of them then broadcast to 100.
To see this you should also rewrite the linear formula:
Y(ij) ~ Normal(b0 + b1*var_one(j) + b2*var_two(j) + b3*var_three(ij) + e(j), e)
e(j) is no different than other linear predictor var_one or var_two, if you treat it as known you can index it in a similar way. Also it avoid the confusion that e(ij) is the same as e(j) that you can index.