Can a system have constant phase response other than +/- 90 deg (and multiples)?
If you're talking about ideal conditions, then, sure, you can have exotic transfer functions with \$\tanh{s^\frac{2}{3}}\$, or what have you. I don't know where this one would be useful, but sure, you can.

If you're talking about real-life cases, even an integrator has a pole somewhere "palpable". With this in mind, filters with -3dB/oct exist (and others):



These are mostly used in noise filtering, this one in particular serves for pink noise, typically filtering white noise (flat) to achieve -10dB/dec, the slope you see measured as ~-10.25dB/dec, because it's in the portion of the phase where it's not quite 45o. It's a lowpass due to the construction, but if you move further right on the slope, you have an integrator with -3dB/oct and 45o. Note that this is more of an idealistic approach, but the response is close to what you might get in real-life.

So, you can make filters that do not obey multiples of 20dB/dec, but the question is what for? That is up to the designer.