You can use simple list comprehension like this,
same_fruits = [frt for frt in fruit if frt in fruit1]
You can as well use data structure in Python known as set, If you are aware of set, it allows you to get intersection.
fruit = set(['apple', 'pear', 'peach', 'orange'])
fruit1 = set(['pear', 'orange', 'banana', 'grapes'])
same_fruits = fruit.intersection(fruit1)
Update (Problem in your code)
Your code does not work because, str type itself is a iterable in Python. So, when you use + operator with list and apple or orange, which is again a utterable, Python tries to extend instead of using append method. You could also make a small change in your code, to make sure that str is appended as a whole instead of treating it as individual components by exclusively using append method. Instead of + operator which in this case is extending iterable with another iterable , use append method.
fruit = ['apple', 'pear', 'peach', 'orange']
fruit1 = ['pear', 'orange', 'banana', 'grapes']
same = []
for f in fruit:
for ff in fruit1:
if f == ff:
same.append(f)
print(same)