I'm a UK Amazon customer who bought an Echo Dot a few months ago. The deal of £3.99 per month for Amazon Music Unlimited on just that device is a plausible fit for my needs, and a friend had asked me to help sort out some themed music for a New Year's Eve party. I knew lending the Dot to another household was plausible because I'd just taken it on a Christmas visit elsewhere. On that occasion I'd even got my host to log in with their own Amazon Prime account to access their music library.
So on the afternoon of 31st December I logged said previous host out of the Alexa app, signed in myself, then asked Alexa to "try Amazon Music Unlimited". (Henceforth I'll just write that I asked to try AMU.) It reported my free trial had started. A few quick test requests worked really well.
So I took the Dot to the party, told it about Wifi and bluetooth speakers. No luck accessing the 40 million songs there though, and asking again to try AMU got me "Sorry, I'm having trouble." Ah well. Odd spot for the Dot I guessed.
On return home, though, same situation. No access to the music, no evidence of a subscription having started. Asking to try AMU, same report. In the history, I see "SystemFaultMusic.MembershipIntentM.
I'm on holiday from work, but I'm still a customer, so I contacted customer services. The Alexa specialist support told me that the subscription process on the device uses Voice Purchasing, and that this simply doesn't work for non-Prime members.
(It remains enabled and selected in my Alexa app, though. And of course I really did successfully subscribe when sure I was signed in as my non-Prime account. As you may have guessed the free trial had been started instead on the previously logged Prime account - my host wisely cancelled it when they got an unexpected welcome email. I will be adding correspondence to a trouble ticket when I return to work I think. But let's just get to the workaround.)
This is the workaround customer service advised: non-Prime amazon.co.uk customers who want to try Amazon Music Unlimited at £3.99 on their Alexa device should instead start a trial of the full service from the website. Once the trial expires you can ask Alexa to downgrade to the £3.99 level.
This may hold for non-UK Amazon too: I don't know. And a proper fix (rather than weird error and support workaround) is in progress, of course.