There are a few different types of Spotify paid accounts. To get the free Google Home Mini, you have to be a subscriber to the Spotify Premium for Family plan.
The Premium Family plan costs $14.99/month, but it allows up to five people to access Spotify Premium (no ads, offline listening, etc.) using individual accounts. That's just $5 more than the usual price of a single Spotify Premium account — it's a no-brainer if even just two people are using the service, let alone five.