How can I create a column with counts between a column and a list in Pandas?

Another solution using Series.str.count:

df['score'] = df['answer'].str.count('|'.join(correct_list))

[out]

             answer  score
0  cats, dogs, pigs      2
1        cats, dogs      2
2        dogs, pigs      1
3              cats      1
4              pigs      0

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Update

As pointed out by @PrinceFrancis, if catsdogs shouldn't be counted as 2, then you can change your regex pattern to suit:

df = pd.DataFrame({'answer': ['cats, dogs, pigs', 'cats, dogs', 'dogs, pigs', 'cats', 'pigs', 'catsdogs']})

pat = '|'.join([fr'\b{x}\b' for x in correct_list])
df['score'] = df['answer'].str.count(pat)

[out]

             answer  score
0  cats, dogs, pigs      2
1        cats, dogs      2
2        dogs, pigs      1
3              cats      1
4              pigs      0
5          catsdogs      0

We can also use Series.explode:

df['score']=df['answer'].str.split(', ').explode().isin(correct_list).groupby(level=0).sum()
print(df)
             answer  score
0  cats, dogs, pigs    2.0
1        cats, dogs    2.0
2        dogs, pigs    1.0
3              cats    1.0
4              pigs    0.0

You could do as follows

correct_list = ['cats','dogs']
df['score'] = df['answer'].str.split(', ')
df['score'] = df['score'].apply(lambda x: sum(el in x for el in correct_list))
df

It will give you the following result

    answer  score
0   cats,dogs,pigs  2
1   cats,dogs   2
2   dogs,pigs   1
3   cats    1
4   pigs    0