Is -3dB bandwidth defined for any type of transfer function?
Maybe you do not realize what -3dB actually means?
-3 dB means that the amplitude of the voltage (current as well) has dropped by a factor \$\sqrt2\$.
That means that the power (= voltage * current) has dropped a factor \$\sqrt2\$ * \$\sqrt2\$ = \$2\$.
And that is the point: the signal power at the output has dropped by a factor of 2 (compared to the passband).
This happens at (a) certain frequency / frequencies and those frequencies are all -3 dB points with which we can define a certain bandwidth.
So yes, the -3dB bandwidth can be defined for any type/shape of transfer function.
"Is -3dB bandwidth defined for any type of transfer function?"
No - in general, it is not. Of course, you CAN define the -3dB bandwidth for any transfer function - if it makes sense. However, for some filter functions (other than Butterworth or Thomson-Bessel response) it is common and agreed practice not to use the 3dB point for defing the bandwidth.
Example: Chebyshev or Cauer (elliptical) responses. In these cases, the end of the passband is defined by the ripple within the passband (when the magnitude in case of a lowpass crosses the value for DC to the last time). For highpass responses this specification applies to the magnitude for very large frequencies.
Because bandpass and bandstop filters can be seen as a lowpass-highpass combination, the above applies to these transfer functions as well.
You can choose the bandwidth to be anywhere your requirements ask for it: -3 dB (typical Butterworth and 1st order), -0.1 dB (typical 0.1 dB ripple Chebyshev), +0.5 dB (unusual, but it can be if ripple exists and is higher than 0.5 dB, while the passband has 1+δ, instead of the usual 1-δ), -40 dB (e.g. unscaled inverse Chebyshev), etc. It's not you who defines that, it's the requirements, so if your filter needs to have a bandwidth at -1.23 dB then that's where it will be considered.
In the examples above, I said that -3 dB is typical for Butterworth and 1st order filters. This is true, at least mathematically. In practise, even then exists frequency scaling, which is why custom bandwidths can be chosen (like the -1.23 dB).
Therefore your question:
is it still valid to say bandwidth of the circuit is ...?
is not the right one unless your requirements specify that the bandwidth needs to be there.
To show what I meant, here's a 2nd order Cauer/elliptic with 1 dB ripple and 20 dB attenuation, and the possible points where the bandwidth can be chosen:
E are in the ripple zone, which you won't see in practice, but it can be done. Point
C is anywhere in the transition region, which can be met in practice, though I probably have too many fingers at one hand for how many cases there are. Point
D is right at the stopband point -- unscaled inverse Chebyshev filters default there -- and it can be a requirement in practice (e.g. the minimum end for the highest harmonic, passband doesn't matter much, only external harmonics). The most common point (for filters with ripple in the passband) is
B, or the end of the ripple, while
C (for points from -6 dB, up) is most common in filters without ripple in the passband.