It is not clear how you calculate correlations using profile projections.

Did you mean the correlation between the variables instead of the correlation
between the means ?

You want the correlation between the width of the variables. Please define width.

In general, correlation between f(x1) and f(x2) could be obtained by performing an error propagation in which things will appear like stdev(x1), stdev(x2) and core(,x1,x2).

As far as I know, the profile histogram gives the mean value of Y and its error for each bin in X (link).
Therefore I want to ask if is there any way to get the std dev of Y for each bin in X.

In case you are interested in the correlation between Y values for bin x_{i} and x_{i+1} just scan the tree gathering the sets { Y(x_{i}) } and { Y(x_{i+1}) } and calculate the correlation coefficient