In theory APT installs latest version of the package available. When I add ppa:elementary-os/os-patches in my 20.04.1 system, I'd be getting "updates" for following packages:
base-files/focal 11ubuntu5.2+elementary9~ubuntu20.04.1 amd64 [upgradable from: 11ubuntu5.2]
gir1.2-gtksource-4/focal 4.6.0-1+elementary2~ubuntu6.0.1 amd64 [upgradable from: 4.6.0-1]
libgtksourceview-4-0/focal 4.6.0-1+elementary2~ubuntu6.0.1 amd64 [upgradable from: 4.6.0-1]
libgtksourceview-4-common/focal,focal 4.6.0-1+elementary2~ubuntu6.0.1 all [upgradable from: 4.6.0-1]
lsb-base/focal,focal 11.1.0ubuntu2+elementary2~ubuntu6.0.1 all [upgradable from: 11.1.0ubuntu2]
lsb-release/focal,focal 11.1.0ubuntu2+elementary2~ubuntu6.0.1 all [upgradable from: 11.1.0ubuntu2]
This packages are essential for elementaryOS and the repository's description itself states:
elementary OS specific patches
This PPA is essentially required by elementary OS
Though not mentioned or warned, this repository may mix up packages on Ubuntu.
A similar mess and mixup can also be caused if you use Pop!_OS' PPA. So, whenever you add PPA/repository, look out for the packages they offer. See if those packages are distro specific and would replace the official packages, don't use it and if you use, don't upgrade or dist-upgrade the system. Your system may report other distribution (for example, Updated Ubuntu 16.04, now I'm running Kali?) and even break the system horribly and would land in dependency hell (for example, if you use Kali or Debian repository).
Now coming to your questions:
What exactly caused my mess (I guess sudo apt dist-upgrade is to blame) and what has been changed (like /etc/lsb-release)?
Yes. dist-upgrade alone shouldn't be blamed. The combination with that PPA should be held responsible.
I don't know what has been changed but I can say above packages would have been surely replaced.
Is there a way (beside reinstalling) to cleanup the issues completely? (I just saw that files like /etc/issue and /etc/issue.net are also changed)
ppa-purge is the best way if you don't want to clean install the system. If it fails, you've to do work, a lot of work, i.e., manually check all packages the replace with the official ones. You can always check which package provide "that" file using
dpkg -S file
and check if you can install (or "downgrade") the official packages without breaking other packages.