You can take advantage of Prime Stations to get relatively close to what you want.
Prime Stations are ad-free, personalised streaming music stations you can use to discover songs from the Prime Music catalogue. You can like and dislike songs and skip as many songs as you like.
If you ask for a station from a particular artist:
Artist stations will play songs similar to the artist you have chosen, along with some of their best songs.
So you should be able to just ask:
Alexa, play an [artist name] station from Amazon Music.
You can also personalise any station by telling Alexa "thumbs up", "I like this", "I don't like this", etc. This will be used in the personalisation process as advertised so you should be more likely to get songs that you do like.
Be aware that you can only play stations for artists that already have a station assigned to them. Spotify can play a station based on any artist whereas Amazon can only play a station based on specific artists.
E.g. "Alexa, play a Coldplay station" would work.
"Alexa, play a Greta Van Fleet station" would not (unless Greta Van Fleet has a station assigned to it by Amazon...).