12v to 5v for a vintage radio project
It would work, but not with your power supply. The echo dot uses a 5.2V 1.8A supply. With perfect conversion, that is 9.36 Watts from your 12V 1A or 12W power supply.

Your amplifier is a max 100 Watt stereo amp  and the Amazon page recommends a 3A supply. Not sure what speakers you are using, so we can't say how much power you are actually using.

In both cases, you don't have enough power to drive both the echo and the amp from that one supply. Easiest thing is to run a second power supply for the echo dot. The extra cable can be velcroed to the other one to prevent clutter. If not, get a bigger 12V supply and honestly any car 12V to 5V 2A adapter to power the echo dot.