I don't know about replacing Make with Docker, but I use the two together to good effect. One of my favorite hacks is adding a 'docker-%' rule in my Makefile to run make commands in a Docker image[1]. It's a bit mind-bending, and there's a few gotchas, but it works surprisingly well for simple rules.
[1]: https://github.com/kevin-hanselman/dud/blob/e98de8fcdf7ad564...