Would these two photos be identical (i.e. depth of field, perspective etc.)?

• Depth of field: not exactly, but very similar if you also multiply the f-number used by the same crop factor. If you use f/2.8 with your 50mm lens on an APS-C camera, you would use f/4.48 on the 80mm lens with a FF camera to get the same overall DoF. You might have the same amount of DoF, assuming both images will be viewed at the same display size and from the same distance, but the distribution in front of and behind the actual focus distance would be slightly different. On the other hand, if you use f/1.4 with the 80mm lens on a FF camera, you would need a 50mm lens at f/0.875 on the 1.6X crop body to get the same DoF.
• Perspective: Yes, as long as the camera position is identical, the perspective is the same regardless of focal length, sensor size, or the resulting field of view. Perspective is determined by one thing only: The position of the optical center of the camera relative to the various objects in the scene.
• Geometric distortion: No. Some people confuse perspective and geometric distortion, but they are two entirely different things. The first is determined by the position of the camera relative to the scene. The latter is determined by the lens' projection of a three-dimensional world onto a two-dimensional recording medium. Even two different designs of 50mm lenses for the same sensor size can use different projection models and have different distortion characteristics.
• Background blur: The number and shape of aperture blades have a profound influence on the appearance of background (and foreground) blur. We call the aesthetic qualities of the background blur bokeh. It is influenced by the lens' optical design, including how much correction is done for several of the classic aberrations presented any time a lens with real thickness is used to refract light of more than one wavelength. It's also influenced by the shape and number of aperture blades that may be in the optical path. Two lenses used on the same camera with the same focal length and very similar optical formulae can render out of focus areas differently and their bokeh will look very different if their aperture irises are made differently.
• ISO, shutter time, etc.: No. If you change the aperture to get the same DoF, you need to change one of the other exposure parameters to maintain the same exposure. Either shutter time or ISO or some of both must be changed to compensate.

In the end, there's no such thing as exact equivalence when using different sized sensors. In addition to the points above, DoF equivalence breaks down at macro distances¹ and when computing hyperfocal distances (which are based on DoF and how much of the DoF is in front of and how much is behind the actual point of focus).

¹ As lenses are focused to very close "macro" distances in order to give magnification approaching 1:1, their effective focal lengths change enough that using the "stated" focal length of a lens non longer accurately computes the depth of field one can expect.