I will limit my observations to loudspeakers for this response...
Indeed, as Andrew Jones said recently, almost all recordings have manufactured information, they are not the replicas of the exact music space. So, listening for accuracy is a very elusive quest at best. Thus, the No1 prerequisite to establish for yourself whether the speakers are "higher fidelity" than the ones you already use, is 1. be very familiar with recordings, and 2. be even more familiar with acoustic live events. (I mean symphonic concerts, with proper instruments and acoustics - a live event at a bar is although still live but can have a terrible sound nonetheless, as room acoustics is very far from optimal.)
All in all though, we seem to corner ourselves into a dead end when singling out a single parameter and trying to elevate it at the expense of others. (Resolution, frequency range, precise imaging...) A balanced loudspeaker needs to handle all aspects, not one falling suspiciously short. It’s a much happier combination than overdoing one and falling short on another. For example having resolution but not at the expense of tonality, and having frequency range but not at the expense of dynamic range.
When all is balanced, then I can find a loudspeaker that I can live with on the long run.
So what does all that have to do with listening to new speakers? Well, watch out for the balance in the sound. Listen to what is the strength that stands out! That’s also a warning sign that you need to inspect that it’s not at the cost of another parameter. For example, if textural resolution is razor-sharp, make sure that it was not done at the expense of tonality and timbre.
First, just let the speakers break in properly. I just play them without expectations for a month, and listen to random things, including streaming movies...
Often, playing non-audiophile records and material will tell more about the character of the speakers - as special audiophile recordings were made to sound special, and they will so on any loudspeaker. Yet, for me it is vital that I can play any recording. Sure, they will not sound stellar, but they all should sound better than I ever heard them before. And that’s all I ask for.
I’m interested in the musical heritage of all mankind. Playing the most diverse material on a loudspeaker for a few months will establish precisely and accurately whether I want to live with those speakers or not.
More precisely: I never met a pair of speakers that did all or failed all. All of them have a special place in my heart, and I know which to pull for a specific experience. They are singers, and each singer has her favorite song that nobody can follow just the same way.