Technically you're not his girlfriend yet. The whole point of dating is to figure out whether you two are compatible or not. After a few dates, he or you would bring up the topic of being "exclusive." If you both agree to it, then you would be his girlfriend, and he your boyfriend.
Dating is something that is done between two people who express an interest in each other. It's not an automatic relationship. If you enjoyed the date and you want to continue and he also wants to see you again, eventually the possibility of seeing each other exclusively may come up. At that point, if you both agree to it, you are in a relationship (i. e. boyfriend/girlfriend). So in short, you likely are not his girlfriend as far as he is concerned.