Buddha said he is Buddha. Did Buddha carried a false sense of self like you and me?
The term 'Buddha' means 'awakened'. At that time that the Buddha was speaking, the word was not the honorific that it is today. He literally meant that he had awakened to understand the nature of reality; the term was merely descriptive, and did not take on the role of being an identity until some time later.
Of course, there's a tricky aspect to this. The factual condition of awakening to the true nature of reality makes one buddha whether or not one proclaims it, but it is easy for the ego to convince itself that it has 'awakened,' leading one to cling to an identity of 'being buddha.' Obviously one who clings to an identity isn't awakened, but that is often a difficult and subtle discrimination to make. We take it as read that Gautama was using the term in the descriptive mode — as part of his effort to lead people to their own awakening — and not succumbing to attachments in his own right.