is there a way to calculate Matching Distance for categorical variables?

I have tried Gower's Distance to measure the distance. Though the code is working, however I realized that the approach is not feasible as this algorithm uses the concept of Manhattan distance for continuous variables and dice distance for measuring similarity between Binary variables. Hence, I would need to carry out one-hot encoding for all the categorical variables (with >2 categories), i.e. convert all those into dummy variables.

However I do not want to follow the one-hot encoding approach. I am trying an alternative approach which I am not able to research. Is there a better algorithm?

Below is the dataset:

{'Sex': {0: 'Female',
  1: 'Female',
  2: 'Female',
  3: 'Female',
  4: 'Male',
  5: 'Male',
  6: 'Female',
  7: 'Female',
  8: 'Female',
  9: 'Male',
  10: 'Male',
  11: 'Female',
  12: 'Female',
  13: 'Female',
  14: 'Female'},
 'City': {0: 'Lagos',
  1: 'Lagos',
  2: 'Tokyo',
  3: 'London',
  4: 'Tokyo',
  5: 'Lagos',
  6: 'Cairo',
  7: 'Lagos',
  8: 'Lagos',
  9: 'Tokyo',
  10: 'Houston',
  11: 'New York',
  12: 'Lagos',
  13: 'Paris',
  14: 'Chicago'},
 'Region': {0: 'Africa',
  1: 'Africa',
  2: 'Asia Pacific',
  3: 'Europe',
  4: 'Asia Pacific',
  5: 'Africa',
  6: 'Africa',
  7: 'Africa',
  8: 'Africa',
  9: 'Asia Pacific',
  10: 'Americas',
  11: 'Americas',
  12: 'Africa',
  13: 'Europe',
  14: 'Americas'},
 'Graduated': {0: 'No',
  1: 'Yes',
  2: 'Yes',
  3: 'No',
  4: 'Yes',
  5: 'Yes',
  6: 'Yes',
  7: 'No',
  8: 'Yes',
  9: 'Yes',
  10: 'No',
  11: 'Yes',
  12: 'No',
  13: 'Yes',
  14: 'No'}}

And the code I implemented:

import pandas as pd
import numpy as np
import gower

df=pd.read_csv("k_mode_data.csv")

distance_matrix = gower.gower_matrix(df)

cat_df = pd.DataFrame(distance_ma

trix)