In principle you could build a browser that - to a good approximation, you will, for example, realistically not be able to prevent timing leaks - does not leak any information about the system it is running on, just isolate it from the host system. With more effort you could let details about the host system leak into the browser but not out into the network. Well, some information has to flow from the host system into the browser and out over the network, for example key strokes and mouse movements, otherwise you could not do much with this browser.
So you could still try to fingerprint users by their choice of words, typing patterns and things like that, maybe made a bit harder by filtering out high frequency timing information. But at least one could no longer simply fingerprint the host system. Which would again be a trade-off, your screen resolution is not only good for a few bits of a fingerprint but can also be legitimately used to serve content at a proper resolution.