d_test = {
    'name' : ['South Beach', 'Dog', 'Bird', 'Ant', 'Big Dog', 'Beach', 'Dear', 'Cat'],
    'cluster_number' : [1, 2, 3, 3, 2, 1, 4, 2]
}
df_test = pd.DataFrame(d_test)

I want to identify similar names in name column if those names belong to one cluster number and create unique id for them. For example South Beach and Beach belong to cluster number 1 and their similarity score is pretty high. So we associate it with unique id, say 1. Next cluster is number 2 and three entities from name column belong to this cluster: Dog, Big Dog and Cat. Dog and Big Dog have high similarity score and their unique id will be, say 2. For Cat unique id will be, say 3. And so on.

# pip install thefuzz
from thefuzz import fuzz

d_test = {
    'name' : ['South Beach', 'Dog', 'Bird', 'Ant', 'Big Dog', 'Beach', 'Dear', 'Cat'],
    'cluster_number' : [1, 2, 3, 3, 2, 1, 4, 2]
}

df_test = pd.DataFrame(d_test)

df_test['id'] = 0

i = 1
for index, row in df_test.iterrows():
    for index_, row_ in df_test.iterrows():
        if row['cluster_number'] == row_['cluster_number'] and row_['id'] == 0:
            if fuzz.ratio(row['name'], row_['name']) > 50:
                df_test.loc[index_,'id'] = int(i)
                is_i_used = True
    if is_i_used == True:
        i += 1
        is_i_used = False
                           

    name        cluster_number id
0   South Beach 1              1
1   Dog         2              2
2   Bird        3              3
3   Ant         3              4
4   Big Dog     2              2
5   Beach       1              1
6   Dear        4              5
7   Cat         2              6

Note, for Cat we got id as 6 but it is fine because it is unique anyway.

While algorithm above works for test data I am not able to use it for real data that I have (about 1 million rows) and I am trying to understand how to vectorize the code and get rid of two for-loops.

d_test = {
    'name' : ['South Beach', 'Dog', 'Bird', 'Ant', 'Big Dog', 'Beach', 'Dear', 'Cat', 'Fish', 'Dry Fish'],
    'cluster_number' : [1, 2, 3, 3, 2, 1, 4, 2, 2, 2]
}
df_test = pd.DataFrame(d_test)

I want to identify similar names in name column if those names belong to one cluster number and create unique id for them. For example South Beach and Beach belong to cluster number 1 and their similarity score is pretty high. So we associate it with unique id, say 1. Next cluster is number 2 and three entities from name column belong to this cluster: Dog, Big Dog, Cat, 'Fish' and 'Dry Fish'. Dog and Big Dog have high similarity score and their unique id will be, say 2. For Cat unique id will be, say 3. Finally for 'Fish' and 'Dry Fish' unique id will be, say 4. And so on.

# pip install thefuzz
from thefuzz import fuzz

df_test = df_test.sort_values(['cluster_number', 'name'])
df_test.reset_index(drop=True, inplace=True)

df_test['id'] = 0

i = 1
for index, row in df_test.iterrows():
    row_ = row
    index_ = index

    while index_ < len(df_test) and df_test.loc[index, 'cluster_number'] == df_test.loc[index_, 'cluster_number'] and df_test.loc[index_, 'id'] == 0:
        if row['name'] == df_test.loc[index_, 'name'] or fuzz.ratio(row['name'], df_test.loc[index_, 'name']) > 50:
            df_test.loc[index_,'id'] = i
            is_i_used = True
        index_ += 1

    if is_i_used == True:
        i += 1
        is_i_used = False
                           

    name         cluster_number  id
0   Beach               1        1
1   South Beach         1        1
2   Big Dog             2        2
3   Cat                 2        3
4   Dog                 2        2
5   Dry Fish            2        4
6   Fish                2        4
7   Ant                 3        5
8   Bird                3        6
9   Dear                4        7

Computation runs for 210 seconds for dataframe with 1 million rows where in average each cluster has about 10 rows and max cluster size is about 200 rows. I am trying to understand how to vectorize the code.