Identification and authentication are quite different things.
I've never heard of anyone using SSN last 4 for identication; it would likely have duplicates for any group over about 100 people, and most computerized systems, at least, deal with substantially more than that.
Full SSN is a fairly good identifier, as it was designed to be, at least for US citizens and (legal) residents. It isn't perfect, and no system of its size is likely to be: there have been cases of the same number assigned to multiple people by mistake, and rather more cases of people improperly using (i.e. stealing) someone else's number; and there are relatively few cases where one person is assigned multiple numbers over time (but not concurrently). For some applications -- particularly anything related to taxes, such as employment and investment, and certain other government benefits -- it is required to be used to identify persons. (Although Medicare recently finally began shifting to a different, Medicare-only 'Beneficiary Identifier'.) For many others it isn't required, and there is a valid policy debate whether it is better to have fewer multiply-used identifiers or many/all separate and distinct ones. It is certainly easier for people to remember one id, or at most two or three, compared to ten or twenty or fifty, and it makes it possible to (accurately!) link different records, relationships, and parts of a person's life, which is sometimes beneficial and sometimes not.
OTOH, too many systems use as an authenticator either full or partial SSN. It was NOT designed for that, and as Anders correctly said, is VERY BAD for it.