When your oriented vertically, your camera’s physical image sensor is positioned in portrait orientation - and the camera aspect-ratio will match. By contrast, when the iPad is positioned horizontally, the camera sensor is oriented in Landscape.
In Landscape, your gaze will appear to be to be to one side or other of the screen. What you are seeing is a fundamental limitation of tablet and camera design - which becomes more pronounced as screens become larger.
By necessity, the iPad camera is off-axis; the bigger the screen and the closer you are to it, the more this physical limitation becomes apparent. This effect is not limited to iPad, but similarly affects all computing platforms that employ a front-facing camera.
Unless you are looking directly at the camera, you gaze will never appear to be looking in its direction. Ultra wide-angle lenses, such as that employed by the 2021 iPad Pro, seem to visually exaggerate this effect - hence the behaviour that you observe. As you move away from the camera, the off-axis parallax effect rapidly diminishes.
When comparing with a a desktop or laptop PC, the camera is most often placed at the top as opposed to the side of the screen (as is the case when using your iPad in Landscape orientation). As such, when using a PC, you gaze may appear to be below and away from the camera axis. A correctly positioned laptop will have the top-edge of the screen (and hence the camera) will be approximately at eye-level. Whilst your gaze will still be directed downwards, the off-axis effect will be minimised.
It follows that the only way that this limitation could be fully resolved is to place the camera lens dead-centre - looking through your screen. Clearly, this is not a viable proposition with current screen technology.
In conclusion, there is nothing that you can do - other than to place the screen slightly further away to reduce the unavoidable effect. A software update cannot change the physical properties of the camera or the orientation of the imaging chip therein.
NB: It is perhaps worthwhile to note that the new iPad10, available from 26th October, has moved the camera from the top (short) edge of the iPad to the long-edge of the iPad. When oriented in Landscape, with the camera at the top (long edge), the camera now better replicates both aspect ratio and positioning of laptop from-facing cameras.
