You can create a file type for .jl files, and associate it with the editor of your choice. I don’t know what desktop Linux Mint uses, so the way to do it will depend on that. If it’s KDE Plasma, you can do it like this, using System Settings → Applications → File Associations.
I am on the cinnamon desktop. I am unable to locate a menu similar to the one you are showing. The closest I can find is “Preferred Applications”, which only offers a set of drop-down menus to change the default app for exsisting file-types. So I guess my question becomes: How to I add a new file type text/jl in Linux Mint Cinnamon?
There were already mime files present for c++, python, and rust, despite me never doing this manual process for any of those file types. This indicates to me that this process should be automated during installation of Julia.
To that end, I shamelessly ping @davidanthoff and @StefanKarpinski as two people with the power and know-how to make this happen during Julia installation.
Those files are not included in any system package but they are autogenerated from /usr/share/mime/packages/freedesktop.org.xml which is in the Debian/Ubuntu package shared-mime-info. That package already contains info in C++, Python and other programming languages, so it would be best to try and get that file updated by the maintainers rather than creating an output file for Julia. I think the right place to ask (or submit pull requests) is xdg / shared-mime-info · GitLab .
I also checked right now. It was added 5 months ago, so the update probably just has not had the time to arrive to your distro. As far as I understand updates have to go through Debian → Ubuntu → Mint, so it might take a while.
EDIT: and probably the first step is Xdg including it in their next release of the source package. Their latest release is 2.2, released one year ago, and does not include Julia yet.