Why does $\sqrt{\frac km}$ represent angular velocity and not frequency?
"Angular velocity" can be used inter-changably with "angular frequency", but you want to distinguish clearly between those and "cyclic frequency" which is the thing usual just termed "frequency".
The angular quantities are measured in radians per second, while the cyclic frequency is "cycles per second" AKA hertz (Hz).
The "angle" here is not obvious the first time you see it. There are two equivalent ways to understand it at first:
Just accept the math as a guide The $\omega$ appears in the argument to sines and cosines when you write down the time evolution of position, velocity, etc, so $\omega t$ represents a angle and must have angular units.
The "reference circle" To explain SHO in a class where the students don't have calculus we consider an object in uniform circular motion and then project that motion into one-dimension.1 In this view there is an actual angular velocity around an actual circle, but we're just using it to show that SHO is equivalent to the behavior of uniform circular motion along a single axis.
You might object that neither is very satisfying--one purely abstract and the other referring to a teaching/calculating aid and not to objective reality--and to my mind you'd be right.
A more fundamental explanation comes from considering the system in Hamiltonian phase space where its path is a circle2 and has angular frequency $\omega$.
And yes, radians are formally dimensionless so you can write angular frequency as just $\mathrm{s}^{-1}$. But you'll hate yourself in the morning.
1 I actually use an object on a rotating table and a spot-light to demonstrate this construction: the shadow of the object executes 1D SHO on the wall. I've seen a demo where that was paired with a pendulum that could be set swinging next to the wall so you can see that the swing and the slide of the shadow really do correspond, but I never got that rigged for my own classes.
2 The phase space is an abstract space (position, momentum). To get a circular path for the SHO you must use suitably scaled coordinates, but they are also the ones that make the system's energy proportional to the square of the vector in that space and related to the construct that we use to get the ladder operators for the SHO in quantum mechanics and so on. It's all very elegant, but you aren't tooled up to follow the details when people first tell you that $\sqrt{k/m}$ is the "angular velocity".