Disclaimer: I'm not sure if this question belongs here, and I can take it down if it doesn't. I'm just not sure where it should fit, so suggestions are welcome.
By drawing a series of lines on a flat piece of paper, I can convey to you in a very concrete sense any solid object, from a simple cube to an entire cathedral. So, a being that observes a 2-dimensional surface from a 3-dimensional space can infer 3-dimensional objects from that surface.
However, a 1-dimensional surface is not sufficient to represent the simplest 2-dimensional objects, even when observed from a 3-dimensional space. Imagine a taut string stretching across a room. By marking portions of this string with a marker, there is no way I can convey to you the idea of a square, circle, or triangle*. If the 1-dimensional surface were observed from a 2-dimensional space (flatland), it seems that this claim would still hold.
To take it slightly further, imagine as a thought experiment a system that allows one to draw on a 3-dimensional surface (a hologram, for example). Could a being living in four dimensions use this system to represent 4-dimensional objects?
More generally, is there any fundamental relationship between the dimension of a surface, the dimension it is viewed from, and the maximum dimension of objects that can be represented on it?
*Unless we have some arbitrary code that we have agreed upon in advance, but this is not the spirit of my question
You are right that this is not entirely mathematical. Indeed it is more physiological or psychological than mathematical. Never-the-less, I'll attempt to answer what I can.
First you are mistaken that you can make simple line drawings of three dimensional objects and generally have them recognized. This has been studied. What little I know comes from a Scientific American article I read about 20 years ago. The researchers showed various drawings that most people of the world would instantly recognize and interpret as 3-dimensional to members of "primitive" people groups, where "primitive" here means (and only means) having very little exposure to the world's major cultures. These people are just as intelligent as one would expect to find anywhere, but have not been raised and trained in the same memes and other cognants as we have since birth.
What they found is that often these people were not able to interpret 3D line drawings as we do, and the more isolated the person was, the less likely they would interpret them "correctly". Interpreting line drawings as 3D objects is not an innate talent, but something our brains are trained up in during our formative years. We learn to interpret subtle clues such as relative size and positioning. These people are not confounded by many of the optical illusions we are so familiar with, because to them, they are just markings on paper. It is obvious to them that those shapes are the same size because their brain hasn't been trained to re-interpret them as being differently sized.
The flip side of this is that a 2D creature could have similar brain-training and learn to interpret relative spacing of points as indication of 2D dimensionality, perhaps based on an assumption of regularity of the imaged polygon. However, for such 1D drawings, I suspect shading and coloration would be even more important to them than it is to us. (And as someone who works regularly with graphics software with both shaded solid mode and wireframe mode, I can tell you it is extremely important to us. Wire frames are difficult to accurately interpret even for simple solids.) The novel Flatland actually discusses this to some depth, as lowly isoceles triangles were not allowed to paint themselves and pretend to be high-class circles.
That said, you are right that the amount of shape information communicable on a line is considerably less than is communicable on a plane, which in turn is less than can be communicated in space. (I don't mean this in a precise mathematical way, where I'm not sure if the concept I'm after can even be appropriately defined.) Presumably, this trend would continue into higher dimensions.