I have the Spotify app open on my iMac desktop and an Amazon Tap speaker on the same wifi network. But why won’t the Spotify auto-detect the speaker? Found the answer at Spotify Support

Connecting to a device using Spotify Connect

If you’re playing to another phone, tablet, or computer, make sure all devices are logged into the same Spotify account.

Bottom line, there are 2 requirements:

  1. You must have a premium account
  2. You have to be logged into the same Spotify account. And, apparently being on the same family account doesn’t count as being on the same Spotify account, so not everybody in my family can control the speaker.

But this doesn’t quite make sense, because my son is able to play to the speaker on Spotify and he’s using a family-connected account. Will have to troubleshoot some more..