It's a more abstract kind of dimension. The vector is only one dimensional because it has a factor of one unit vector (only one associated "direction". A Bivector has two factors of unit vectors to describe a 2 dimensional region.

That's independent of the dimension of the space it's embedded in. So you can have a one dimensional vector in 2space, 3space, nspace etc. But it only ever describes one direction, an oriented line.

(x, y) are the components of a vector expressed in a 2D space, where space as the medium the vector is in. The dimension of the vector and the dimension of the space it's in are different. You can have a 0-dimension point on the 2D space, a 1-dimension line on the 2D space, and a 2-dimension plane on the 2D space.

In a 3D space, a vector would be (x,y,z), in 4D (x,y,z,w), etc.

"One dimensional" simply means a vector is a straight line, as opposed to a plane or a solid.

geometrically, a vector is direction that points from one point to another -- it is one-dimensional, like a line between the two points.