i think that there can't be a motion in the horizontal direction with constant velocity when there is an acceleration in the downward direction. If the bird flies at an angle then the vertical component of the force will be compensated by mg and still there will be a net horizontal component of the force, so the bird cant move with constant velocity
well if we take the cage out of the problem, i would argue that any bird can do exactly what you say it can't--well maybe not do-do birds or ostriches. Birds in flight are subject to 4 forces, a constant acceleration downward, lift, drag, and thrust. If lift = mg, and thrust = drag, you have constant horizontal flight. I'm not trying to make the question seem foolish as I suspect there is more to it yet.
sure its not as straight forward for a bird as a plane, but mg acts downward thru the center of gravity, lift upwards thru the center of pressure, drag is slowing the horizontal velocity, and thrust (this is tricky) since it is also generated by wings in a rowing kind of motion provides force horizontally.
i think it's not possible in long run. assume the bird flies in a cage with a constant horizontal velocity, then after a while it will hit the grate and therefor change its horizontal velocity to zero. hence, in order for horizontal velocity to be constant it has to be zero all the time, but with zero horizontal velocity it is impossible to fly (unless there is an acceleration acting in the vertically upward direction cancelling bird's weight, which goes against the problem statement). so no, it's not possible.