This occured to me while writing a mathematics/physics library. Given a 3d shape such as a cube, since it provides a simple demonstration, you have three commonly used measures: length, width, and height. All three are measured as distance, the same dimension. If it were to really be three dimensional, wouldn't we be considering something like distance, time, and mass? Furthermore, given a flat drawing such as a square, how can we really say it is 2d, as it could just be a surface draught (as is commonly done in engineering: several faces, and then an isometric).
Or am I completely off, and why?
Yes, a three-dimensional vector space has "directions" in the sense that a vector is directional (a point often made in introductory physics materials to delineate "velocity" from "speed" by saying the former has "direction" that the latter lacks).
However the "three" part of the qualification needs clarification. How many directions are there in three-dimensonal space? The most likely conclusion that a student would draw is that there are infinitely many different directions, left, right, up, down, sideways...!
So we need some mathematical formalism to explain why "three" is special in numbering directions, and this gets to the issue of linear independence. In a more specialized setting (Euclidean geometry) one might use the notion of two directions being at right angles (orthogonal), but strictly speaking this notion of an angle between two vectors is something that comes not with abstract vector spaces but rather with inner product spaces, an extended definition.
In any case in the setting of a simple vector space we can define three-dimensional quite precisely by saying there exists a basis of three vectors, a basis being a set of linearly independent vectors that span the vector space. Perhaps you could set a goal of reworking this formalism in a perferred terminology of directions, but I'm not aware of any true alternative approach to how we count the dimension of a vector space.