He means professional in this case because indeed when there are rumors about him then what must be done as a professional player is to continue to focus on his club rather than doing things that make his team lose.
But indeed in this case actually I personally still don't believe he left Tottenham even though there are Bayern who are ready with good offers and offers still he will still end up at Tottenham in the end.
Harry Kane did well to debunk the rumors and speculation linking him to Bayern Munich why he is still working hard for Tottenham, that is what is expected of any professional footballer to do. I am also not sure that Kane is interested in going to Bayern Munich, we should not forget that Kane is an England boy and England players love staying around England football clubs especially when they are good players. It is easier for Kane to move from Tottenham to Manchester United than from Tottenham to Bayern Munich .