Already been done on a tiny bike.
It must do it entirely by balance on the bicycle seat, the same way a human would. It can't have propellors, reaction wheels, parts in contact with the ground, or other such cheating. It does not need to be humanoid.
It must be able to ride any normal human bicycle; the bike can't be specially designed for it. (It's ok if it fails on a more esoteric design or a bike with broken parts.) It only needs to ride it on a flat surface, but it needs to be able to do so for at least 30 seconds. It must be able to pedal forwards at >10mph.
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This is definitely possible with modern robotics, but it’s difficult and I don’t think it’s likely to happen.
If you were to task me with building this robot, I would cheat by include a spinning counterweight inside the robot, which could be spun in order to generate leftwards or rightwards force. But this would not satisfy the question criteria as I read it. The robot should balance by turning the handlebars and going forward, or (less likely) by pushing off the petals and leaning.
@MatthiasPortzel For example, this robot has a reaction wheel in its chest which allows it to balance without moving forward, or ride forward on a balance beam, something a human wouldn’t be able to do. https://youtu.be/G3_0OzaoQ00
My no position is contingent on this being considered cheating.
@IsaacKing Humans can lean left and right, but that's different than what a robot with a reaction wheel does. The reaction wheel turning one way produces a turning force in the opposite direction. Humans do this too, but it manifests as pinwheeling our arms when falling (or turning when unicycling—when a unicyler wants to turn clockwise, they make small counterclockwise circles in the air with their arm). It's a different form of balancing from the form of balance that humans use when biking, which is primarily turning the handlebars left and right.
If reaction wheels are allowed, it's a much easier problem because balance is basically free. There's a ton of exiting work on using reaction wheels to balance top-heavy objects that are stationary (look up "self-balancing" on Youtube). Steering is then simple too, you can't just lock the handlebars, but you only need a little bit of feedback to keep you going straight. This is what's shown in the video that I linked before. That's a human-sized robot effortlessly riding a bike on a balance beam because the reaction wheel handles balancing. Pedaling is difficult still, especially if you want the robot to work with different models of bike without having the feet manually strapped in. But it's really a mechanical challenge, not a difficult software one.
If reaction wheels aren't allowed, then the robot has to ride the bike like a human. In order to balance, it has to be moving forward and it has to adjust the handlebars to point in the direction that its leaning in order to keep the bike underneath it. Pedaling is still a difficult mechanical problem, and a slightly higher stakes one, since the robot can't balance unless it can pedal. In a quick scan of Youtube I didn't see any robots doing anything like this, except for the small robot that you linked above.
@MatthiasPortzel You're right, that's my bad. I removed that qualification since it didn't match the title.