Snowbird writes:

First, motorcycles don't necessarily move in two dimensions in a turn.

Neither do aircraft.

Second, you might also want to ponder what the motorcycle driver vs. the
pilot is looking at.

The driver of any vehicle needs to look where the vehicle is going.

Maybe the bike rider wants to look at the intended track of his bike on the
road, in order to spot any bumps.

No, that results in target fixation, and accidents.

Or maybe he just wants to optimize his
turn to the available width of the road and looks at it more broadly. In
either case it would seem to make sense that the rider's sight perception
improves, if he tilts his eyes more parallel to the road.

Motorcycle riders look where they want to go, just like pilots.

The pilot, on the other hand, does not look at any road ahead. He's
interested in the nose vs. horizon sight picture as well as the instruments.
That's a different case and it's not self-evident that tilting the head
parallel to the horizon would improve the pilot's turn performance. On the
contrary, especially if the pilot uses the VSI and altimeter to maintain
altitude, it's probably easier to read them with the eyes level relative to
the instrument panel.

This is a VFR pilot?

Third, it just might be possible that the pilot's stereoscopic vision can
better help him maintain altitude in the turn by visual cues, if he keeps
his head still.

Stereoscopic vision ceases to be a factor beyond around 10 metres, so it is
never important in the air--which is why full-motion simulators use
collimation to make everything seem infinitely far away, without bothering to
simulate 3D.

Fourth, a bike rider leans forward, while a pilot leans back in his seat.
Can have impact on how the head turns in a turn.

I was taught not to lean in any direction that isn't aligned with the bike.

I'm not aware of scientific proof of the above, but neither of the reverse.
So until the opposite is credibly shown, I'll contend that a motorcycle is
different from an airplane.

Different in many ways, but very much the same in turns.