Yes and no. Theoretically, yes you may use one database for all services. Historically it was even an old assumption and some services still ask you for a table prefix to allow various services to use the same db without clashes. However it's no more recommended and you should absolutely avoid going this way.

But nothing blocks you to have several separate databases on the same server. Here your mysql docker container play the role of a database server. You can use the same container for different services using different databases. Create one specific user/database per service inside your container (you may wrap the official mysql image inside your own image to script its creation), then add it in your docker-compose file and make all your services depend on the same service.

I'd love to have a link to the wallpaper :)