Ok I think I found where to access the Account Setting Page. For my instance, I use two different AppleIDs for iCloud services and Media/Purchase (iTunes/App Store). My current device is a MacBook Pro 16" (2019) running macOS 14.3.1. The instructions I list are for my setup and might vary if you have an older OS etc.
1) Open "System Settings"
2) Click on iCloud / AppleID at the top of the left menu
3) Click "Media/Purchases"
4) Click the "Manage" button on the "Account" line. There is also a "Manage" for Subscriptions but that is a different than "Account Settings"
5) This pulls up the "Account Setting" which we are discussing from the Music App, but instead, it is accessed through the AppStore. It seems to be the same menu and options as the previous Music App "Account Settings".
It could be a glitch in the Music App, but my guess is Apple will be removing accessing Account Settings from the Music app in the future as they completely decentralize previous iTunes services into multiple Apps for all OS's. Windows is the last iTunes holdout, but it will soon (2024) be replaced with 3 apps for iPhone syncing and media purchase: Apple Music, AppleTV, & Apple Devices (instead of Finder/AppStore for macOS).
Hope this helps!