On macOS this is more of a “session” feature and I know Finder, Preview, Terminal and Safari support it. Though most web browsers, even WebPositive, maintain the state of tabs at least across shutdown of the app.
Right now in Haiku the only thing which really maintains a session state across shutdown is Tracker. The Terminal session state in macOS (which even maintains previous terminal output) is very nice and I would like to see that added to the Haiku Terminal.
Overall this is generally an application specific feature, though maybe the OS can provide some helper functions to make it easier. But the good news is that initially it should not require anything special in Haiku.
If there are applications you always use and want to start at boot-up they can be added as shortcuts to the user startup folder at /home/config/settings/boot/launch/
. Or you can add to the UserBootscript
(which also is what launches the programs in the startup folder.) This is the source file though I forget offhand where it ends up on the installed system, but you can search for it in Tracker: haiku/UserBootscript at master · haiku/haiku · GitHub
If you want the exact same window setup of an application which does not maintain that itself, you could always set up some hey
scripts to be run in the UserBootscript to move things around.