Does the concept of damping apply to 1st order systems?
Damping means the real part of the complex conjugate pole is larger or smaller, relative to the magnitude of the pole. To be fair, it means much more, but mathematically that's where it gets to. A 1st order system has only real poles, for example:
$$H(s)=\frac{1}{s+1}\quad\Rightarrow\quad s=-1$$
There can be 2nd order systems, and higher, with real poles, and in those cases, damping no longer applies:
$$H(s)=\frac{1}{s^2+3s+2}=\frac{1}{(s+1)(s+2)}\quad\Rightarrow\quad s_1=-1\,,\,s_2=-2$$
But for a 2nd order system, or higher, where there are complex conjugate poles, the realpart is (considering a unity magnitude) \$\Re(s)=-\sqrt{1-\Im(s)^2}\$. With this you get the usual pictures.
In short: you don't see the forest because of the trees, i.e. you got stuck in the namings and can't see the meaning of damping.