You guys are getting way too specific and complicated. I'm sure the answer is much simpler:

We say "get on the plane" because passenger aircraft fill the same role, historically, as passenger ships, and we always said "get on the boat" in the past.

Why we used "on" instead of "in" for ships in the past is an entirely different question, but when you're talking about today's usage, it's because of that very historical precedent.

The car->carriage thing was already pointed out. "In" is used in that case for the same reason: Historical precedent.