Please provide the definition of a degree that is the ratio of two quantities that have units of length.I believe that degree is also dimensionless. It is also defined as the ratio of two quantities that have units of length.
Also degree and radian are related by a factor. So if radian is dimensionless then degree also is.
The only definition of the degree that I am aware of is that the angle swept by a full circle is 360°. That is very analogous to saying that the distance covered between the equator and the pole is 40 million meters.