It's because both radians and degrees are are a ratio of a length to another length and are thus dimensionless. No matter how you measure it, all angles are without a unit.

It's most obvious with radians but it's also the case with degrees.

Using radians, you are guaranteed to not introduce unusual extra terms to rescale angles, if you use any other scale of angle you will have to keep track of extra terms.

That may be useful in whatever you're doing. I work in degrees quite often and I'm careful to keep track of the 2pi/360 terms that crop up all over the place. With grade measure you have to keep track of 2pi/400 terms and with turns you have to keep track of 2pi/1 terms that will repeatedly show up.

Again, depending on what you're doing, this may or may not make sense to do.

In general, mathematics works out easier when the scaling term is 2pi/2pi because then you have a lovely 1 scale factor you don't have to keep track of.

Agreed. Though sometimes it's useful to keep track of 'fake' units like for angles, to make something like dimensional analysis work for you.

But that's more for analysis of your code / formulas than when you actually go and compute things.