What does it mean for a unit vector to have a magnitude of 1?

Imagine a Cartesian coordinate system whose origin is associated with two unit vectors, ê and â, in a 2D-space. Now, let 0.5 cm be the unit of length in this coordinate system.

The magnitude of a unit vector is, by definition, 1; Does this mean that the magnitude of one of our unit vectors is 1 unit of length? Or, in other words, does this mean that $$\left | â \right |$$ = $$\left | ê \right |$$ = 0.5cm?

The basis vectors are dimensionless quantities with magnitude 1. You create dimensional vectors to represent positions, velocities, accelerations, forces, etc. by multiplying each basis vector by a dimensional scalar and then adding together. For example,

$$\mathbf{r}=(2\,\text{cm})\hat{\mathbf{e}}+(3\,\text{cm})\hat{\mathbf{a}}$$

In other words, the components of a vector carry its dimensions. That way, the same basis vectors can be used to represent all kinds of different vectorial physical quantities.

You have defined a unit of length to be equal to $$0.5\,\rm cm$$ which we can call $$1\,\rm ash$$.

Suppose you move $$5\,\rm ash$$ in the $$\hat e$$ direction.

The displacement is $$5\,\rm ash\, \hat e$$ which is $$2.5\,\rm cm\, \hat e$$

This means that $$|\hat e| =1$$ irrespective of what you are describing and whatever the units that you are using.