Linux distribution package managers work like this. Updates can run in the background, processes pick up new libraries whenever they are restarted, but updates can happen before that during normal operation mostly without any disturbance (there are some exceptions to this, e.g. Firefox behaves strangely if updated while running). Unix file systems generally allow overwriting files in use, processes that already had it open get the old version, new processes get the new file transparently. Only kernel updates really need a reboot, and the occasional related stuff like kernel modules (but even that is rare, you can e.g. update your Nvidia driver and only need to restart X11, not the kernel, however, it is easier to just reboot). There is e.g. checkrestart and similar tools that tell you which processes you might want to restart after a given update by checking library dependencies.

There are occasional rare glitches this system wouldn't catch, like e.g. artwork being changed resulting in a mix of old and new icons, but that doesn't happen if you just stick to normal releases of stable distributions like Debian.

I always pity my windows colleagues in the week after patch tuesday. Our Linux boxes are automatically updated 4 times a day, reboot is automated for login servers for when nobody is logged in (with some help of loadbalancing), other servers generate a mail and we reboot them when convenient. But reboots are rare. Automatic unattended updates breaking stuff is even more rare, I think two instances in 10 years.