This incident, as Leacock describes it, must have looked more than a little irregular if viewed from an outside perspective. He asks to see the manager in private, suggesting himself to be a person of some significance to the bank, while he also (in his general anxiety) finds himself struggling with some of the most basic tasks of banking. One can expect this behavior to draw a fair amount of attention (and speculation) from the bank's other customers.
That being said, we don't actually know whether the other people in the bank believe him to be an "invalid millionaire" (of if they're really thinking all that deeply about him at all). When the bank manager presumes he is a Pinkerton, we have confirmation of that assumption, because the bank manager says it out loud. In the case of the other people in the bank, however, all we really have is the speculation of Leacock's narrator, who has already been established as an overly anxious, emotionally overwhelmed individual who is very much out of his comfort zone.
He sees the others in the bank and draws assumptions as to the impression he must be making, but we only see his perspective on this experience, not the perspective of anyone else in the bank.