If you've recently migrated your stack to a fully docker containerised setup, docker-compose can be a great replacement for an old vagrant provisioning setup, as the startup time will usually be a fraction of that needed by running vagrant up. It won't make sense for every project, but it's definitely a viable option for some.
The equivalent tool is docker-machine, not docker compose.
Though they de-emphasised that tool in favour of Docker for Mac and Docker for Windows which interact directly with the platform hypervisor to create a Linux VM.
You can get close but as someone else said, they are mean't to be processes. You can get more control with Vagrant for a lot of things and have actual VMs.