Yes motorcycles do lean into the wind by themselves I know that experentially.
I knew that gyroscopic precession was involved but didn't understand what frame geometry had to do with it. I found this
explanation most helpful.
"Yes, most bikes are self-correcting. A gust causes the steering axis to translate away from the wind side. (The gyro effect of the rear wheel tends to restrain the bike's rolling more than it prevents the frame from yawing, because of the leverages involved. The frame yaws, and the steering head translates.) The contact patch, due to trail, is behind the extension of the steering axis. Therefore, in a crosswind from the left, the steering axis moves slightly to the right. The contact patch is then slightly to the left of the steering axis, meaning that the bike is steering itself to the right. This banks the bike to the left, into the crosswind.
You can take your hands off the bars entirely, and this feature works just fine."
This makes sense to me as I think about what happens in a crosswind. There is an initial slight movement downwind followed by the bike leaning into the wind. Basically the bike is counterstreering itself.
"Here's an interesting photo showing how a bike automatically leans into a 50 mph side wind blast coming out of the fans in the building at the left"