The way that it’s mathematically expressed, space actually comprises a set of three 2D planes, arranged orthogonally. In this usage, each individual “dimension” is actually a plane, or a set of two perpendicular directions, where each direction is also called a “dimension.” The x and y dimensions combine to form the xy-plane which is perpendicular (normal) to the z axis; it is the direction of the z axis and not the location of the z axis that is the thing which is perpendicular to the x direction and the y direction. For these reasons, it would be more meaningful (less confusing or less arbitrary) to specify our mathematical treatment of space as 2D3 rather than 3D. It’s the reason why our mathematical treatment of Euclidean 3-space has octants… 2x2x2=8.
There exists a deeper understanding of what a 2D plane actually is, though. It’s most commonly defined as two lengths (or axes) perpendicular to one another, sure, but what does that really mean? The concept of orthogonality brings the concept of relative direction (or orientation) into the picture. This structure of relative orientation, when combined with length, is the defining characteristic of a two-dimensional plane, and the defining characteristic of Euclidean 3-space.
There is also another use of dimension where it means a particular quantity that can express a distinct physical phenomenon or characteristic. In this usage, the meaning of dimension is analogous to base quantity in physics, and is the foundation for the analytical technique known as dimensional analysis.