With speakers it matters for the amount of feedback that can be applied. Feedback is needed to make some drivers play really low. It keeps distortion down but that doesn't always sound right with everything. It can be what you need for "slam" sometimes. One way to approximate feedback is the output impedance of an amplifier.
When it comes to power delivery it's not like that at all. There's no feedback.
There's no such thing as "fast" or "instantaneous" power. It's a perpetuated nonsense myth (even I use to think it was a thing). What happens if you don't have adequate power is voltage drops and causes a very noticeable flatness in sound. People think they're hearing a lack of current, but unless you have minuscule wire, like smaller than you can buy or anyone would EVER use, then you've got current. The issue is that if there's too much heat from current then voltage will drop. How many times has a powercord felt hot to you? Never. But it's not unusual to come across a piece of equipment with an inadequate transformer or capacitance. Again what you hear is voltage dropping. With capacitors it's because they release energy, or, voltage and current. The faster they release current the faster voltage goes down, so you need a lot of capacitance to keep voltage adequate. Transformers are a little more complicated, but basically they can only pass so much current before you get heat and loss of voltage. There's soooooo many ways to affect sound, and it's easy to just claim it's current if you're a manufacturer...
There are other things that can cause flops in voltage. You can have a bad resonation or oscillation that'll eat up so much current that you get loss of voltage. It can sometimes happen for only very short moments of time.
How much are you willing to pay to measure impedance? Good measuring devices are probably $10k... You might be able to get something that can do line level voltage in the hundreds of dollars range. They have to specifically do impedance.