min find the minimum value of an iterable with respect to a function. min(lst, key=func) in Python can be roughly considered as (ignoring details like calling f as little as possible or walking through the list only once)
retval = lst[0]
for item in lst:
if func(item) < func(retval):
retval = item
return retval
In your case, it finds the item v in updated_unique_list such that len(set(u) ^ set(v)) is minimum.
Now, ^ between two set is their symmetric difference. set(u) ^ set(v) returns a set of elements that does not appear in both u and v. The size of the result will be zero if u and v are equal, and is maximum if they do not overlap at all.
Therefore, the function tries to find an object in the updated_unique_list which has the most common elements with u.
If you want the "normal function" just expand the min according to the algorithm I've described above.