That makes sense to me. There was a specific thing I was looking for in my original question which was to understand how related vs. arbitrary the shared term

That makes sense to me. There was a specific thing I was looking for in my original question which was to understand how related vs. arbitrary the shared term 'dimension' is. I think I found an answer after browsing a few Wikipedia articles—but happy to be corrected if it doesn't sound right: The usual notion of dimension that people think of is topological only; fractal dimension (i.e. Hausdorf dimension) is also concerned with metric properties. These metric properties are what creates the need/possibility for fractional measures. In that case, the scenario I was considering with 'fractional axes/directions' is not implied by the fractal dimension concept: fractal dimension is dealing with new subject matter, not just an extension of the old concept to deal with real numbers.