Dimensionality of 1D line through a 2D plane

Question

What is the dimensionality of the 2D plane with a 1D line going perpendicular through it as depicted here? Could one come up with a coordinate system that describes the entirety of this space? In general, how is the dimensionality of a shape formally defined in geometry and how does one go about in finding it? Also, how is the embedding of an $N$-dimensional object in a higher dimensional space formally defined?