this is correct: local state for webapps is stored in the browser, even if you host them locally
i don't think this is easier to sync multi-device than eg gossip's local files

nostrudel does store its settings in a relay-side 30078 event as well, this gives it automatic synchronization of settings between devices, maybe this is enough for what you want 

 yeah, this is also one of the big hassles with web apps is the web browser doesn't easily let you connect the app to a local cache relay... you could then also connect that cache to wireguard and use the same cache on your other devices as well

to do shared state like this you have to build a separate configuration/cache service