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.