A person gets 'on a bus'.
'into' has the sense of being 'inside and surrounded, enclosed inside something'
This might seem just what happens when one boards a bus.
However, the first trains and buses were not the fully enclosed vehicles they are today, and would have open sides, and the top deck might not have a roof. So 'getting on' this 'platform on wheels' seemed appropriate at the time...and has just continued as the expression till today.