PERSPECTIVE IS ALWAYS ABOUT CAMERA POSITION RELATIVE TO THE SUBJECTS VISIBLE IN THE FIELD OF VIEW. IT IS NEVER ABOUT LENS, SENSOR SIZE, OR ANYTHING ELSE.
As far as I understand, to photograph the same image on the APS-C
camera as the Super 35 camera you need to use a wider lens, with the
focal length needed being determined by the crop factor. Using a wider
lens though, this would affect how the depth is perceived e.g.
background objects appear further away than they do when using a
longer lens. So this would not result in the ‘same’ image. Is that
NO IT IS NOT. IF THE CAMERA IS IN THE SAME POSITION THEY HAVE THE SAME PERSPECTIVE. PERIOD.¹
Parts of the scene that are visible in one image will be visible in the other. Parts of the scene hidden by other objects in the scene will still be hidden by those other objects. This is because the lines from the camera to each object are always the same and are always straight. What may change is the way those objects are projected by the lens which will affect the shape of those objects as projected onto the focal plane, but the same parts of each object will be visible and the same parts will not be visible as long as the camera and the objects are all in the same position. That is what perspective is!
Any difference between two images made with different lenses shot from the same position and with the same field of view will NOT be due to differences in perspective. They will be due to the differences between the two lenses in terms of geometric projection caused by the way each lens refracts the light passing through it, by the different absolute resolution of the two lens/camera systems, by the difference in color and light transmission between the two camera/lens systems, etc. But there will be no difference in perspective if both photos are taken from the exact same position.
The difference between an image taken with an 8mm fisheye and an image taken with an 8mm rectilinear lens is not a difference in perspective if both images are taken from the same shooting position. It is a difference in the geometry of projecting a 3D world onto a 2D sensor (or film). If there are two objects there with part of one object in front of part of the other object, as long as you shoot from the exact same spot the exact same parts of the rear object will be seen and the exact same parts of the rear object will be hidden by the nearer object. That is what perspective is!
The difference in how those two objects are shaped differently in the two photos is not one of perspective, it is one of geometry of projection. Two totally different things. Perspective distortion is a result of shooting position. Geometric distortion is a result of the different ways different lenses project the same perspective onto a flat imaging sensor or piece of film.
For more, please see this answer to a different but related question: Is there a difference between taking a far shot on a 50mm lens and a close shot on a 35mm lens?
See also this answer to: Does wide angle equivalent in crop sensor skew image?
Perspective is dependent only upon shooting distance - both the distance from the camera to the subject and the distance from the camera to the foreground/background, and the ratios between them all. If you shoot from the same distance with two different sensor sizes or different focal lengths and crop the wider shot to match the narrower angle of view perspective is identical.
If you shoot from the same position with both cameras, then taking the Super 35 camera and crop it to the same angle of view of the APS-C sensor that will give you pretty much the same picture, other than the differences in optical quality between the two sensors and the resolution lost to cropping.
But even if you were to shoot with the same camera, shooting from a different distance will give a different perspective. This is because shooting distance is the only thing that determines perspective. The focal length and sensor size then determine the angle of view and framing from that shooting distance. So backing up with a crop sensor to get the same framing of the subject as a larger sensor would at a closer shooting distance also gives a different perspective: The relative sizes and shapes of items closer and further away from the camera will shift as the ratio of the distances of the various items to the camera changes.
Image copyright 2007 SharkD, licensed CC-BY-SA 3.0
Here's an extreme example of the effect differences in shooting distance have when using different focal lengths to get the same framing from different distances. The change in perspective is due to the change in shooting distance and the different distance ratios between the various elements in the scene and the camera as the camera moves forward and back to preserve framing of the subject at various focal lengths.
How can we ensure an equivalent image is captured, both in terms of field of view and perspective?
Shoot from the exact same position with a lens+camera combination that gives the same angle of view. It's that simple. If you want the same depth of field, also apply the ratio of the sensor sizes/lens focal lengths to the f-number used.
¹ The reason we think a wider focal length causes objects in the background to be smaller than they would be if we had used a longer focal length lens is because when we use a wider lens we move closer to the subject. This changes the ratio of the camera-subject distance and the camera-background distance. If we shoot a subject from 15 feet and the background is another 15 feet behind the subject,the ratio of the distances from the camera is 1:2 (15 feet/30 feet). If we use a wider lens and move to within 5 feet of the subject the ratio is now 1:4 (5 feet/20 feet).
For further related reading, please see:
Why do you need to change your position rather than just focal length to affect perspective?
What is the difference between perspective distortion and barrel or pincushion distortion?
Is there a difference between taking a far shot on a 50mm lens and a close shot on a 35mm lens?
Does wide angle equivalent in crop sensor skew image?
Focal length on Full frames and cropped sensors
Can a telephoto lens have a wide field of view?
How does focal length change perspective?
Why is the background bigger and blurrier in one of these images?
What does it really mean that telephoto lenses "flatten" scenes?
Myth Busting: Focal Length & Perspective