I think you need a scalar product to define volume. But fuzzy thinking an analogy: Think friendship, define three concepts related to friendship and that they are orthogonal, now establish some numeric scale to measure each concept, then you have just defined a volume to measure the degree of friendship. Unfortunately there is no canonical way to establish those orthogonal features, but they could be obtained applying a linear model to big data sets of measures of friendship.
The units of length are whichever you choose them.
You are free to choose different units of length for each of the dimensions of a multi-dimensional space, but that does not bring any benefit and it introduces many constant factors corresponding to the ratios between the different units of length in all mathematical formulae.
Therefore everybody chooses to have the same unit of length for all dimensions, in order to have the simplest formulae.
Of course, this is true for the multidimensional spaces in which a scalar product of the vectors is defined. When there is no scalar product, the units of length for the different dimensions are independent. There is no way to compare them and the question whether they are the same or different is meaningless. In such spaces, angles, areas and volumes are also undefined, only the affine geometry is applicable.
Edit: Just to further the thought. Do all dimensions have the same unit length? In 3D, the spacial x,y,z dimensions have the same unit length. In 4D like 3D+Time, does the t-dimension have the same unit length as the spacial dimensions? What is 1-unit of t meant vs 1-unit of x? What does it mean to say a sphere in 4D with radius r having the same "distance" from the center in all 4 dimensions? What unit length is r in?