As I understand it, the proper length of an object is defined as the length of the object in its rest frame. In terms of the metric it is defined as the length of the spacetime interval between two space-like separated events, i.e. $$dl^{2}=\sqrt{ds^{2}}$$ (with the "mostly plus" signature).
Now, suppose that an observer is at rest in an inertial frame that is itself at rest with respect to a given object that the observer wishes to measure. Why is it the case that one considers the "simultaneous" length of the object, i.e. when $dt=0$, such that $$dl^{2}=\sqrt{dx^{2}+dy^{2}+dz^{2}}$$ Is it simply so that it agrees with the definition of spatial distance in Euclidean geometry or are there other intuitive reasons for why it must be the case (analogous to the definition of proper time in which the proper time of an object is equal to the coordinate time of an observer who is at rest [i.e. $dx=dy=dz=0$] with respect to the object, such that $d\tau=\frac{1}{c}\sqrt{-ds^{2}}=dt$)?!