How to assign a value to a column for every row of pandas dataframe?

You're looking for a cartesian product of both dataframes. One way around this in pandas is to create a common and unique key for both dataframes and perform a merge (any, as there is a complete overlap):

df.assign(key=0).merge(object_raw.assign(key=0), on='key').drop(['key'], axis=1)

   brand_name  category_id object_name
0       Nike           24     T-shirt
1       Nike           45      Shorts
2       Nike           32       Dress
3    Lacoste           24     T-shirt
4    Lacoste           45      Shorts
5    Lacoste           32       Dress
6     Adidas           24     T-shirt
7     Adidas           45      Shorts
8     Adidas           32       Dress

Another way using itertools.product() which gives Cartesian product of input iterables.

import itertools
df=(pd.DataFrame(list(itertools.product(brand_name.brand_name,object_raw.object_name))
             ,columns=['brand_name','object_name']))
df['category_id']=df['object_name'].map(object_raw.set_index('object_name')['category_id'])
print(df)

---

  brand_name object_name  category_id
0       Nike     T-shirt           24
1       Nike      Shorts           45
2       Nike       Dress           32
3    Lacoste     T-shirt           24
4    Lacoste      Shorts           45
5    Lacoste       Dress           32
6     Adidas     T-shirt           24
7     Adidas      Shorts           45
8     Adidas       Dress           32