True, but can’t you cork a Klein bottle just as easily?

I’d ask the inverse. What definition of “inside” can you apply to a traditional bottle–so as to say that a ship is inside the bottle–that could not also be applied to a Klein bottle? Both of them have a single opening that leads to an enclosed, dead-ended volume.

A Klein bottle may only have one surface, and therefore you can argue it has no topological inside. But a traditional bottle is topologically equivalent to a flat disc, so the same logic would say you can’t put a ship inside one of those either.