"* Offer is available for new subscribers who connect an eligible device to an Apple device running iOS 15 or iPadOS 15 or later, for a limited time only. Offer good for 3 months after eligible device pairing. Plan automatically renews at your region’s price per month until cancelled. No purchase necessary for current owners of eligible devices. Service availability varies by region. Restrictions and other terms apply."
These are the restrictions of the 6 month free subscription to Apple Music with a purchase of a HomePod Mini.
You weren't the direct subscriber, but you still were consider subscribed to Apple Music. You can try contacting Apple Support and seeing if it doesn't count, however, I believe you're considered a subscriber since it was shared through Family Sharing.
I hope this answers your question!