It is not assumed that reality has "dimensions" in all cases. Dimensionality is a purely mathematical concept, and whether it applies directly to reality itself depends on whether reality is made of mathematics or not. That particular question is an open question in philosophy.
What does have dimensionality are the scientific models we use to make sense of reality. In these cases, we assert that there is an interpretation of these mathematical models which can be used to make true statements about reality. This is so ubiquitous in our society that we often elect to not draw a line between the two of them. The fact that we don't draw such a line leads to the kinds of questions you ask here.
One key rule, which CriglCragl brings up, is that physicists do classify some of the dimensions in their models as spatial. They do so because we have observed a symmetry: the universe tends to behave the same in each of the directions of these dimensions. The laws of physics do not behave differently if you move two meters forward rather than two meters to the right. They apply the same. Likewise, they do not behave differently if you face forward vs. face right.
Because of this, we do usually assume that the laws of physics operate on some spatial manifold, which is the mathematical term for these sorts of multidimensional spaces. Many of these models have 3 dimensions, the classical 3 dimensions of space. Other models do have more. Relativity operates on a 4 dimensional space consisting of 3 spatial dimensions and 1 time dimension where you can't quite decouple the time dimension like we like to do in classical thinking. You really do have to think of them all together. String theory posits more dimensions (I have heard as many as 11), and uses them to explain some of the behaviors we see geometrically rather than simply assuming they are true. Indeed, if we assume the universe is mathematical, there's nothing to prevent the universe from having infinite dimensions of space, other than that's really awkward to work with for mere mortals (I like to leave the infinite dimensional spaces, like Hilbert Spaces, to the Quantum Mechanics experts!). But as for the dimensional of reality itself, it is not easy to assume that phrase even has a meaning without assuming several interesting statements to go with it.
One key challenge with any higher dimensional theory is to explain why 3d thinking does such a good job at explaining how our world works that we could make it thousands of years before observing the need for additional dimensions. For example, in the case of relativity, it's because we typically never had enough relative velocity with respect to something we could measure to see the relativistic parts creeping in. You could always make a naive 4d theory which assumes everything is an infinite brick in one direction, so that we can ignore the extra dimension, but that kind of structure would quickly decompose under the known laws of physics and chaos theory unless the structure was immaculately perfect. (Immaculately perfect coincidences bug physicists. They don't like them)
Incidentally, it's worth remembering that lower dimensional models are useful in every day life as well. While writing this, I'm using a 2d model to think of this computer screen and the text on it, completely oblivious to the thickness of the LCD materials which are shaping the images for me.