Is there a way to plan and organize your LEGO project like a GitHub hosted software?
While you could indeed manage the project with a tool like GitHub (using the Issues feature), you might be better off initially managing the tasks through a tool like Trello which allows you to create tasks and move them through various status (i.e. Planned, In Progress, Review, Complete) in a fairly simple, intuitive user interface.

An issue with managing the actual project files within source control is around the format of them - some file types work better than others when it comes to source control, and some formats of those types work better than others as well.

Git is a "Distributed Version Control System", which means that anyone can work on any file in the repository on their local machine - this is usually fine when working with text based files, but is not so good when you have binary files, because understanding the differences between two versions is often not easy. If the tool you choose stores projects in binary files, Git is probably not the best option.

I would strongly recommend that you consider a Centralised Version Control System such as Subversion (SVN) or Team Foundation Version Control (TFVC or TFS) as this allows your collaborators to "checkout" or "lock" a specific file to say they are working on it, and then when they've updated it, "check-in" their changes - this will (hopefully) get around the main pain points of trying to merge changes in these files, albeit with the caveat that you might still have people check-out a file and then "go dark" and forget to check it back in for a long time.

Sadly for you, most developers have moved to Git based source control, as it's very good at dealing with code files, and generally much quicker when working locally, allows for "offline" working, etc. and solves the problem of people working on different parts of the same file or project simultaneously, so most of the free options (GitHub, GitLab, etc.) focus on that. You might be able to use the free options within Azure DevOps to create a TFS based project, or find another SVN host for your project. Most of these also offer "Work Item" tracking for task planning and management.

In terms of the current software for building your MoC, the likely contenders are:


Stud.io - uses a binary (.zip) format. This doesn't play well in Git. Furthermore it's a password protected archive, so you can't easily extract the underlying .ldr files.
LDraw - uses a text based format. Potentially a better fit for Git, however the format itself would be very easy to mess up in merges if two people work on the same model files at the same time.
LEGO Digital Designer - uses a text (XML) based format. Potentially a good fit for Git. While XML can get messed up in merges, it's usually easier to see in merge tools where the changes should belong, and so easier to correct/adjust if two people work on the same model file. However, this is likely end-of-life now, especially with the announced purchase of BrickLink by The LEGO Group.  


  What about Studio and LEGO Digital Designer – what will happen with the
  two digital building experiences since they’re quite similar?
  The LEGO Group is planning to support and evolve Studio and will review opportunities
  for both services following the closing of the deal.