You do not indicate whether you have exchanged emails or calls in person with this supervisor. (At the very least, my email program will remember who I've interacted with even if I don't, and I confess that this has at times been my first checkpoint.) However, I suspect that your dealings thus far have been with a different person and that you and the supervisor have simply been assigned to one another. However, s/he did sign the letter. It is quite possible that they may not remember this as their PA gives them a pile of letters to sign everyday (but then it would be unprofessional for them to plead complete ignorance, if they have a PA with whom they can check what happened!). There could also be a bog standard mix-up.
In any case, be forewarned that this mess is a sign of things to come. You will be supervised by an overworked junior staff member or postdoc, your supervisor will rarely if ever see you and most likely still not have the slightest clue who you are when the internship is done, unless you deliver them a high-impact manuscript they can put their name to.