Apple Music Library not showing up on Apple TV

Please disclose which model Apple TV you have, as there may be relevant differences between models.

Identify your Apple TV model - Apple Support

On Apple TV (4th generation or later) with current software tvOS 12.1, the Computers app should show six menus: Film, TV Programmes, Music, Photos, More, and Search. The More menu should contain Home Videos, Podcasts, and Audiobooks.

Apple TV (3rd generation and earlier) do not support Apple Music, except via AirPlay from another device.

On Apple TV (4th generation or later), with an Apple Music subscription and iCloud Music Library activated, you should probably use the Music app instead of Home Sharing, as that app is tailored for that subscription.

The Music app on Apple TV (4th generation and newer) is for subscribers to Apple Music, iTunes Match, and for iTunes Store purchases. Its Library is the iCloud Music Library as part of the Apple Music or iTunes Match subscription.

To listen to music from your Home Sharing iTunes Library, instead use the Computers app on Apple TV to connect to the Library, then go the Music menu option to see and play your music. (Home Sharing needs to be activated on both the computer with iTunes, as well as the Apple TV.)