Can be done in your Sound System Setting. There is a option to specifically turn the music for mounts off.
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Because the system isn't built for it. It's sadly that simple. And if they could already have given you a pick for BGM selection, I should hope they'd double down and just let you pick a mount theme, whichever you wanted. Could even be things saved locally so as to not burden the servers with saving that info.
The new theme for Red Hare would be Horizons Calling by Nobuo Uematsu, it was actually an old FFXIV version 1.0 track (it was the theme that played when you rode on the Vesper Bay to Limsa ferry). Ironically it was one of the very first pieces of FFXIV music I ever heard, as SE had it playing by default on the old FFXIV 1.0 teaser website back in 2009 when FFXIV 1.0 was first announced. But then I also like Borderless too so it figures I guess. ;)
So what music does each of these play now?
The Red Hare
Horizons Calling?
Nezha Chariot
???
Indigo Whale
???
Aquamarine Carbuncle
???
Citrine Carbuncle
???
Mystic Panda
???
The Red Hare
Horizon's Calling
Nezha Chariot
Horizon's Calling
Indigo Whale
Merry Wanderer's Waltz
Aquamarine Carbuncle
Merry Wanderer's Waltz
Citrine Carbuncle
Merry Wanderer's Waltz
Mystic Panda
Merry Wanderer's Waltz
Does it actually do that, though? I swear I remember turning my music off way back when I got tired of the chocobo theme, and all I got was silence and zone noises. My big complaint was that I couldn't turn my mount music off and just listen to the background music... but if I was wrong/they changed it, I'll be very happy. I prefer some zone music to some mount music.