coefficient of overlapping (OVL) for two distributions

Question

I would like to calculate the overlap between two distributions for which I have samples but not the PDF.

If I had the PDF of my two variables f and g, then the OVL would be $∫min(f(x),g(x))dx$. However, I do not have the PDFs, I have arrays of samples that were drawn from unknown PDFs.

So, taking my starting point to be two arrays arr1 and arr2, I have coded the following in Python:

import numpy as np
def OVL_two_random_arr(arr1, arr2, number_bins):
    # Determine the range over which the integration will occur
    min_value = np.min((arr1.min(), arr2.min()))
    max_value = np.min((arr1.max(), arr2.max()))
    # Determine the bin width
    bin_width = (max_value-min_value)/number_bins
    #For each bin, find min frequency
    lower_bound = min_value #Lower bound of the first bin is the min_value of both arrays
    min_arr = np.empty(number_bins) #Array that will collect the min frequency in each bin
    for b in range(number_bins):
        higher_bound = lower_bound + bin_width #Set the higher bound for the bin
        #Determine the share of samples in the interval
        freq_arr1 = np.ma.masked_where((arr1<lower_bound)|(arr1>=higher_bound), arr1).count()/len(arr1)
        freq_arr2 = np.ma.masked_where((arr2<lower_bound)|(arr2>=higher_bound), arr2).count()/len(arr2)
        #Conserve the lower frequency
        min_arr[b] = np.min((freq_arr1, freq_arr2))
        lower_bound = higher_bound #To move to the next range
    return min_arr.sum()    

I have validated this function with some known OVL results. For example, @wolfgang proposes here some R code that calculates an OVL of 0.6099 for two normal distributions with the following properties N1(mu=1, sd=1), N2(mu=2, sd=2).

When I use my function, it yields the following results:

arr1 = np.random.lognormal(loc=1, scale=1, size = 100000)
arr2 = np.random.lognormal(loc=2, scale=2, size = 100000)
OVL_two_random_arr(arr1, arr2, 100)

==>0.65658

OVL_two_random_arr(arr1, arr2, 1000)

==>0.61107

OVL_two_random_arr(arr1, arr2, 10000)

==>0.607089

In other words, the size of the bins I use influence the results.
What should I do to avoid this? Is there a way to estimate the best number of bins (the same way we have Scott's rule and Silverman's rule for KDE)? Is there a way to avoid setting the number of bins completely?