Apple Music Family subscription not recognized for family member

Re: Apple Music: Family Sharing group with my mom

Apple Music Family subscription doesn’t seem to apply to her account.

If needed: Add you mom to your shared subscriptions:

How to: Add a family member to your shared subscriptions - Apple Support

You mom may need to check that she is signed in to her iPhone, iPad, or Mac, with her correct Apple Account she will use for Apple Music: If needed see Apple Support's article (link below):

How to: Sign in to your Apple Account on the web or in your device settings - Apple Support

Else if your mom uses an Android phone:

See: Use Family Sharing with Apple Music on Android - Apple Support

Seems that your mom will also need to log-in to Apple Music with her Apple Account for Apple Music to recognise her eligibility.

Eg: One way is to:

Open the Apple Music app and tap Listen Now.

In the top-right corner, tap the photo icon.

Sign in with her Apple Account (usually an email address) and password.

Else: It is likely that if she proceeded to start that new subscriptiion process when asked,

then Apple Music would at some point recognize she is already eligible.

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Re: "Is there a way to connect her to Apple Music remotely,

so that she can use the Family subscription without being asked to pay again?"

Seems your mom would only need to log-in, if Family Sharing includes her.

Everyone gets their own library to which they can add whatever music they prefer.

Eg: to get started:

If there are any previous purchases made with her own Apple Account,

these purchases can be re-downloaded,

or

she can add any music available via Family Sharing :

What to do: If you can't find your family's shared apps and content - Apple Support

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If needed: Apple Music have a Support website online: (info. / support / contact options etc.)

Website: Music - Official Apple Support

Sorry Anastasiia Samoilova,

I thought I understood, and addressed those Music issues in my reply.

Re: — the app keeps asking her to set up her own subscription or add a payment method. Is there any way to fix this remotely from my side?

Add you mom to your shared subscriptions:

On your iPhone or iPad, open the Settings app, then tap your name.

1. Tap Subscriptions.
2. Scroll down and make sure Share with Family is turned on. 
3. Tap the Music subscription you want to share.

Then your mom just needs to log-in to Apple Music with her own Apple account to start using her Music Library.

If needed:

Purchase sharing is a feature that you can use after you set up Family Sharing. 

When you turn on purchase sharing, everyone in your family gets access to the apps, music, films, TV shows and books that family members buy from Apple.

Click the links to see:

How to share apps and purchases with Family Sharing - Apple Support

and

How to download apps and content that your family members purchased - Apple Support