Indeed. That's precisely why some browser vendors don't want to give random access to your computer to any and all website. And don't want to give every app and site access to the data that should be private to a particular web site.
From this point of view sandboxing websites seems like a good compromise.
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It's also strange that the argument I keep hearing is "but apps have unrestricted access, so to preserve security and privacy... let's give web sites the same access as native apps". It doesn't work lime this.