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.
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.