Similarity between 2 statistical distributions
One method is Kolmogorov-Smirnov test. Kolmogorov-Smirnov test checks whether two samples are drawn from the same continuous distribution where sample sizes can be different. It's p-value is close to 0 when two samples follow the same distribution and close to 1 when they do not follow the same distribution. So you can use 1 - (p-value) as a similarity metric.
import numpy as np
from scipy.stats import ks_2samp

np.random.seed(52)

n1 = 200
n2 = 300

mu_1 = 5
mu_2 = 5.1

sigma_1 = 0.3
sigma_2 = 0.2


sample_1 = np.random.normal(mu_1, sigma_1, n1)
sample_2 = np.random.normal(mu_2, sigma_2, n2)

result = ks_2samp(sample_1, sample_2)

print(result.pvalue)


1.4998994601889137e-08

Note that there are also other methods such as Bhattacharyya distance, Kullback–Leibler divergence. Some implementations for Kullback-Leibner can be found also here.