Mantis Bug Tracker might be a bit more popular than Phabricator. We know about 3 links to it since March 2021 and only 3 links to Phabricator. We are tracking product recommendations and mentions on various public social media platforms and blogs. They can help you identify which product is more popular and what people think of it.
My company has used Phabricator, which is an alternative to Azure Devops, for about 5 years. A few days ago they announced that the project won't be maintained anymore, so we are looking into alternatives. We have around 50 devs+product folks that use Phabricator at the moment. Source: almost 3 years ago
You also mention Gitlab is close to what you would like - maybe Phabricator is even closer: https://phacility.com/phabricator/. Source: about 3 years ago
Https://phacility.com/phabricator/ I think checks most of those boxes, maybe all. Source: about 3 years ago
I use https://mantisbt.org/ Mantis, no need of Docker. Source: over 1 year ago
Perhaps Mantis Bug Tracker? Admittedly I've never installed/used it myself, but looks to be built using PHP so might work on shared hosting. Source: over 1 year ago
We use MantisBT at work. Simple, just a trouble ticket tracker, no bells and whistles, which is exactly what we wanted (along with a REST API I was able to integrate it into the app so that users don't even have to open the site). Source: about 2 years ago
GitLab - Create, review and deploy code together with GitLab open source git repo management software | GitLab
Jira - The #1 software development tool used by agile teams. Jira Software is built for every member of your software team to plan, track, and release great software.
GitHub - Originally founded as a project to simplify sharing code, GitHub has grown into an application used by over a million people to store over two million code repositories, making GitHub the largest code host in the world.
Redmine - Flexible project management web application
BitBucket - Bitbucket is a free code hosting site for Mercurial and Git. Manage your development with a hosted wiki, issue tracker and source code.
Trac - Trac is a project management and bug/issue tracking system. Provides an interface to Subversion and an integrated wiki.