As far as I understand, VSCode browser can't install browser extensions and doesn't even have Dev Tools. For example, IntelliJ IDEs ship with CEF-based browser that has Chromium Dev Tools, but it still doesn't support browser extensions.
It has a browser built in, which is "just" an iframe, and it definitely doesn't function as a full blown web browser, but it does work for previewing your dev work, and given it's a dev tool it makes sense that the experience would be editor+debugger+tools first, with browser as a second class citizen (rather than chrome with an embedded terminal which would be the other way around).