There is no ambiguity about the clipboard being discussed. Kaiser confirms that Oswald was using the one he found on the 6th floor. He knows it is the same one because he recognized it.
Mr. BALL. And how was it you weren't using it on this day?
Mr. KAISER. You see, when he first started there
Mr. BALL. Who is "he"?
Mr. KAISER. Lee---when he first started to work there he got my clipboard and started using it.
Mr. BALL. Did you give it to him to use?
Mr. KAISER. No, he just picked it up and started using it and I just went and made me another one.
Mr. BALL. You recognized that clipboard when you saw it?
Mr. KAISER. Yes, because my name was all over it.