There are a few cases where loading matters (not "matching").
One is when a piece of equipment has a transformer output.
A transformer is a bandpass filter, which response depends very much on its loading. Particularly at HF, there is a resonance, governed by the leakage inductance and parasitic capacitance. Loading the output helps taming this resonance. It is often done with a simple termination resistor, or with a less simple Zobel network (C and R in series).
When designers feel the need for a termination resistor, they naturally choose 600 ohms, as it's a value that has been sanctuarized by years and years of telegraphy/telephony.
It clearly has nothing to do with efficiency nor power "transfer".
Another example is microphones, in particular dynamic and ribbon types.
Due to their inductive nature, they constitute a natural low-pass filter in conjunction with the load they are connected to.
That's the main reason why current methods recommand bridging them, with about 10x their nominal impedance. Incidentally, it also helps with sensitivity and S/N ratio.
I believe I already mentioned passive loudspeaker x-overs.
Back to the OP, except for passive sources*, matching impedance is not an issue when considering voltage/energy transfer, since any resulting attenuation can be compensated elecronically.
However, proper loading of output stages is sometimes necessary.
*Even though matching could be considered as beneficial in optimizing the connection of passive sources, it turns out that frequency response considerations result in a preferrence for bridging.
Actually, matching the impedance of a ribbon mic or a turntable cartridge is hardly feasible, due to the large variations with frequency.