I've heard we actually see in 2D, not 3D.

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I remember someone saying that if we actually saw in 3D, we'd be able to see all sides of objects.

Our vision is 3 dimensional as we have binocular vision.
Each eye perceives the same scene from a slightly different point of view, our brain laces the information and the differences together to provide us with the perception of depth.
Next time you're in a car (as a passenger not the driver) moving along, cover one eye for a few minutes and see how you go with perceiving depth with vision from only the other eye. You'll find it's more difficult to judge depth and the speed of other vehicles coming towards you.

To see all of the sides of objects around us you'd need eyes on stalks, and even then you can only see what's in front of the eye, you can't see through it.

Technically we see in 2D but our brains interpret that as a 3d image\model of the world.

From an image processing perspective, we do indeed see in 3D, not 2D as your friend surmises.

A linescan camera sees in 1D. If you were to show it a square, it would only see a line.

A regular camera sees in 2D. If you showed it a square, it would see the square. If you showed it a cube, it would see a representation of a cube mapped to a 2D shape. People observe a similar phenomena when they see with only one eye.

Your two eyes together, however, see 3D. While the image from both eyes is 2D, additional information is derived from the slight differences in perspective from each eye, and that is used by the brain to reconstruct the image of the 3D object.

The visual processing abilities of the brain are impressive. In fact cannot even emulate the visual processing abilities of simple brains with our modern computers. Consider that most birds and prey animals do not have binocular vision, yet are still able to construct 3D images by looking at the image with each eye.