Surely, I am normally aware of some specs like power output, THD and, maybe some other basics.
But by knowing specs of a component do you really have an understanding of what a piece sounds like?
No, not really. Specifications are really meant to ensure you aren’t being ripped off. The amplifier measurements are a great example of this. Established by the federal trade commission, I believe, not the NIST.
These ratings are meant to ensure a 100 Watt amplifier really has more output than a 20 Watt amplifier and it’s not just a matter of being supremely optimistic. It also helps you figure out if a speaker is going to be too difficult to drive when you match a speaker's minimum impedance to the amp's.
I know of almost no speaker measurement commonly sold that offers any better. In this case though the minimum impedance and sensitivity ratings can be quite helpful in getting you into the ballpark of being loud enough with your amplifier.
With cartridges and preamplifier impedance ratings it can help you match up one to the other.
The character of a piece of equipment however is very complex.