Let's look at objective measurements of four audiophile-grade amplifiers mentioned in this thread as "musical" (some of them may be not exact model mentioned, yet shall be close enough in "amp family sound"):
https://www.stereophile.com/content/plinius-sa-reference-power-amplifier-measurements
https://www.stereophile.com/content/plinius-sa-100-mkii-power-amplifier-measurements
https://www.stereophile.com/content/leben-cs300-integrated-amplifier-measurements
https://www.soundstagenetwork.com/index.php?option=com_content&view=article&id=1505:bhk-labs-measurements-hegel-music-systems-h360-dac-integrated-amplifier&catid=97&Itemid=154
All of them would be characterized by objectivists as highly distorting. And yet these amplifiers find many buyers who consider them thoroughly enjoyable. At one time or another, this or that authoritative reviewer would call them musical.
There could be many common perceptual phenomena at work, as well as individual hearing system differences. Also, preferred most often listened to music genres, and even specific artists.
It is no different from widely differing preferences in food, wine, closing, cars etc.Yes, there are common qualities that have to be present in all of them, yet it's the differences that make them unique enough for a specific buyer to choose.
I once had a conversation with a man who is both an accomplished audio gear designer and a successful businessman. He keeps multiple authentic vintage audio systems, almost all of them objectively highly distorting.
Why? Because each of the systems is best for reproducing a specific genre of music originating in a specific era. If one wants to hear the artist's intent, he better listen on a system that this artist used to approve the master mix.
So, my take is that "musical" has correlation with objective metrics, yet is heavily influenced by specific individual hearing system peculiarities, and specific gear that specific music piece was recorded and produced with.