Cannot start a 3 month free trial for Apple Music family plan

• I see a VPN icon in your screenshot. I could see how that could cause issues with regional things like Apple Music subscriptions, in combination with some VPN services. Note that regional subscriptions require the Apple ID region (its billings address) to match the region for the device (and the region for its iTunes Store). For a mismatch, you will get signed out for the service. Try setting up without the VPN.
• The Apple Music Family subscription is available in many regions, but not in all regions. See here: Availability of Apple Music and iTunes Match - Apple Support

Did you already set up Family Sharing? Are you the organizer for the Family Sharing group?

If not, then take care of that first.

Family Sharing - Apple Support

Set up Family Sharing - Apple Support

Get an Apple Music Family subscription - Apple Support

I can’t think of other possibilities for that message. I’m sure the fine people at the Apple Store at One Infinite Loop will be happy to help you set it up and/or explain the weird message.