If we need to imagine that a point is dimensionless, how can we accept that a line is one-dimensional.
Have a look at this webpage.
A point as part of a line is a line segment of zero length with dimension of the line.
If L = L1 - L2, L has dimension of L1 & L2 even when L1=L2.
If (x,y,z) has dimensions of the coordinate system, then so does (0,0,0).
One is reminded of a nest where every interval has a dimension, down to the point common to all intervals.