Linux container technology, by which I mean the Linux kernel APIs used by Kubernetes, CoreOS rkt, and Docker, do not have an opinion about whether the processes that run as containers are ephemeral or persistent. Containers can be just as persistent as the host they're running on.
For example, consider a Linux distribution using systemd. Systemd might launch regular processes upon system startup, and it might also launch containers where both types of processes are equally persistent on the parent host. See the documentation for systemd-nspawn [1] and machinectl [2]. The Arch Linux wiki has more details about how to use that distribution with nspawn [3]. Containers managed in this way are just like other system processes aside from being containerized.
[1] https://www.freedesktop.org/software/systemd/man/systemd-nsp...
[2] https://www.freedesktop.org/software/systemd/man/machinectl....