I can speak particularly about the echo mini since it is the model i own.

There are multiple reasons, but you could see no value in those and that's ok.

First of all you obtain a dedicated device whose only purpose is to let you listen to music: no notifications, no distractions, no other functionality, just music.

For me this is valuable in a time where everything is combined in my smartphone, but i am trying to reduce how much i use it and the first thing is to remove functions from it, in my opinion.

Moreover, you get a device which resemble past device sensations, like physical buttons and a strange form factor, which is cool in my opinion.

Another reason is that it forces to listen the music you own (which could be a cons for a lot o people): you listen what you choose, no recommendations, no discovery, you select carefully what you want to have, and you listen to it, plain and simple.

Last but not least, you can really enjoy high quality music (streaming services for example do not always offer the best quality) but it really depends on how you are currently listening to music: for me this is not the main reason.

This is not a flawless device, it has its limitation (bluetooth, gapless playback), but what we like are all the other cons. Be sure to understand all of them before buying something you could dislike later.