Groupby Month and Year and get the count of unique items as columns

This approach works efficiently, only need to get the dummies and count grouping by the date with the desired frequency:

df = pd.DataFrame({'Desc':['Car','Bus','Car','Bike','Car','Car','Car','Bike','Bike','Bus'], 
                   'Date':['2022-04-21','2022-05-31','2022-05-31','2022-05-31','2022-05-31',
                           '2022-06-14','2022-06-19','2022-06-19','2022-07-19','2022-07-19']})
df.Date = pd.to_datetime(df.Date)
df = df.set_index('Date')

pd.get_dummies(df,'','').groupby(pd.Grouper(freq='M')).sum()

result:

Date     Car  Bus  Bike
2022-04   1    0    0
2022-05   2    1    1
2022-06   2    0    2
2022-07   0    1    1

You can try pd.crosstab

df['Date'] = pd.to_datetime(df['Date'])

out = pd.crosstab(
    df['Date'].dt.strftime('%Y-%m'), df['Desc']
).reset_index().rename_axis(columns=None)

print(out)

      Date  Bike  Bus  Car
0  2022-04     0    0    1
1  2022-05     1    1    2
2  2022-06     1    0    2
3  2022-07     1    1    0

You can also do it manipulating the strings without converting them to datetime. First parse them to year, month and day and put them in a pandas DataFrame

df_ix = pd.DataFrame((x for x in df.Date.str.split("-")), columns=["Y", "M", "D"])

Then concatenate these features with you original DataFrame and use them to group by year, month, and vehicle type. Use count to aggregate the results for each group (keep a single column and rename to avoid confusion)

df_multiindex = pd.concat((df, df_ix), axis=1).groupby(["Y", "M", "Desc"]).count()["Date"] 
df_multiindex.name = "Vehicle counts"

The output is

print(df_multiindex) 
Y     M   Desc
2022  04  Car     1
      05  Bike    1
          Bus     1
          Car     2
      06  Bike    1
          Car     2
      07  Bike    1
          Bus     1
Name: Vehicle counts, dtype: int64