No, it’s not Coriolis forces or gyroscopic stability. It’s all down to the front wheel geometry. A ridable bicycle will have the front wheel contact point a short distance behind the point where the straight-line projection of the steering pivot meets the ground.

Such a bike can be pushed by hand, held only by the saddle. As the bike leans, the front wheel steers into the lean providing a stabilising torque.

Here are two good articles. The first is a classic, widely quoted.