Alright, I was able to fix it by installing it under /system/app which doesn't require any extra configuring. I'll share how I did this in an edit of the post. Thanks for all the help :)

It is Signal specific, all other apps, atleast on my phone, that use cloud messaging work.

Since most people that I know use messaging apps that nag about missing GMS I decided to get MicroG. Now, I want Signal under the Cloud Messaging hood mostly due to battery saving.

Just tried it with Aurora and I get the same message

Oh, I installed MicroG as an user app. I'll try installing it as a system app as a last resort, since reinstalling every messaging would be really tedious.

Also, one question. What do you mean by not installing it as a Magisk module? I'm not sure what other way there is, other than directly writing to /system

Maybe. I installed it as an user app.

How should I install it as a system app if not from a Magisk module?

I already fixed it

Just so you know Firejail is a setuid root binary with a lot of code. This means that if that code gets exploited and the sandboxed process escapes the sandbox it will have root privileges instead of the running users.

Sources: https://github.com/netblue30/firejail, https://madaidans-insecurities.github.io/linux.html#firejail

I reinstalled Signal from the website apk and got the same message. I currently have 5 apps registered, all displayed in the companion app.

Yes, however, it won't prevent the process launching as root because firejail is setuid. This means that the person executing the file transitions into the privileges of the owner of the file, in this case, root. Since this root process remains in the background it can theoretically be exploited by the sandboxed process that firejail spawns.

The odds of this happening to someone who isn't getting targeted are very very low so using firejail is still alright, but you should consider a non-setuid solution like bubblewrap or just a VM.