I wish every browser does that! Web site has no business storing data on my computer more than 4093 bytes (that is already too much) per domain in my computer just because I visited a web page. 10 MB - 10GB of data is too much.
Oddly enough the handful of web apps I’ve known to use local storage don’t communicate this at all. Developers seemingly treat data in browser local storage as existing indefinitely and unlikely to be deleted, when in fact it’s probably the most easily accidentally lost form of storage an app can have.
If, on the other hand, you're explicitly uploading or entering data on a web site, well then yes, the default assumption would certainly be that the data is getting saved on the web site's servers (if it's being saved at all). That's how the web has worked for a very long time, with the exception of cookies and a few other more recent but not widely known storage mechanisms which in most cases aren't considered sufficiently robust for persistent on-device storage.