Imagine a humanoid creature with only 1 eyeball the size of a tennis ball, it has a rare genetic mutation so it had 2 conjoined pupils! I suspect since the 2 pupils are just side by side it would not be able to sense how far objects are relative to each other since the focal point is very short or it brains cannot tell the difference because of the fixed focal point so there is insufficient information to patch the missing puzzles?
Can they perceive depth?
YES. Of course they can. You don't even need the mutation, a single eyeball with a single pupil can perceive depth.
How can we know this?
- Your allowed to drive a car with only one working eye. Judging depth is one of the most critical road skills - needed for everything from following distance estimations at highway speeds to parallel parking.
- Personal experience: I had to wear an eyepatch for 2 weeks because of a medical condition. I had no problem judging depth with only a single eye, even in unfamiliar environments.
- Anecdote: I knew a guy with only one eye and asked him. He responded by challenging me to a game of pool and I barely got a shot in.
- Gamers aren't hopeless: People play games on a computer screen and can judge depth in a 3D world even though there is only a single camera giving information.
- Movies are watchable. Have you ever watched a movie (not in 3D) and been confused about the relative depth of objects in frame?
- It used to be my day job. Using images taken from a single camera but at different times and positions to calculated depth information for a world was literally a source of income for me. The algorithm I worked on was known as a Photogrammetry algorithm.
- Try yourself: Shut one eye, and move your head slightly. The "video" feed your brain got and the accelerometer in your inner ear combine to give depth information from the relative movements of objects in the world. Even works while moving forward.
Are they able to perform stereopsis?
No useful information will be available by the conjoined pupil even though there technically will be two feeds of image. L Dutch has given an answer below which I agree with and have upvoted.
As you correctly suspected, there won't be any different information to infer the depth from.
As you can see from the sketch above, a two eyed viewing system, due to the separation between the two eyes will give a slightly different view of the same object.
A single eye instead, while still giving a small separation between the images due to the non zero distance between the pupils, has the issue of sharing the same retina. How is the brain supposed to tell through which pupil a certain stimulus came?
In the binocular view it knows which is right and which is left, in the monocular it doesn't.