If by "replicating" you mean replicating the experience, then you're looking for Dokku

Easy setup + MIT license and you get the same git push deploys, Heroku-compatible buildpacks or bring your own Dockerfile.

I would recommend running it on probably on Linode / DigitalOcean / Vultr.

Dokku is great for a single host. If you have a more complicated setup you can go a long way with post-receive hooks, although it won't be as magical without buildpacks.

Fly.io comes close. You have docker based builds on the cloud with native multi region support. Postgres is in beta right now.

They also have built-in Heroku migration: https://fly.io/docs/app-guides/speed-up-a-heroku-app/

DigitalOcean's App platform things plays nicely with Heroku buildpacks from what I've seen