(Extending my previous comments into an answer...)
I'm not sure about "most people". I'm Gen X and I'd expect most people from my generation and before think of "chrome" as a bright shiny "silver" or "mirror-like" metal, mostly used on car exterior trim before the '80s but also on other metals that should both look nice and be protected from corrosion, such as taps/faucets. The web browser hasn't even existed for 15 years yet.
Anyway, the era and area when cars had the most chrome plating was 1950s America in the postwar boom. Initially just for corrosion protection on bumpers but went crazy with it by the end of the decade. In the '60s and '70s it went back to just corrosion protection on bumpers, door handles, mirrors etc and had all but disappeared by the beginning of the '80s.
Since I was a child I've had three main interests: language, classic cars (mainly '50s and '60s), and computers. The spot on the Venn Diagram where all three intersect is tiny, but we have found ourselves there (-:
In the automotive world I have only ever known "chrome" to be used in a literal sense. Short for the metal element "chromium". I have never heard of it being extended to mean any other kinds of nonfunctional aesthetic trim, add-ons, flourishes, etc. Not formally and not in slang either. It was extended slightly to cover other "silver" or "mirror-like" polished shiny metal or fake metal. This would include polished stainless steel mouldings and "plastic chrome" in interiors which I believe used evaporated aluminium. It was not even ever used for other colours of shiny metal such as gold or anodized pieces.
To check my suspicions I searched regular dictionaries and then slang dictionaries online. Most only have the literal sense of the metal. Wiktionary and one dictionary of slang also cover the GUI sense. The Routledge Dictionary of Modern American Slang and Unconventional English, page 200:
They give the date of 1991 as the first use they found and note that it was in another dictionary, Eric S Raymond's The New Hacker's Dictionary. In lexicography, sourcing a word only from another dictionary is not always seen to be as good as sourcing it from primary materials "in the wild". According to Wikipedia, the New Hacker's Dictionary is derived from the Jargon File, which is already mentioned in another answer here, and with a slightly earlier citation for "chrome" in the sense we're investigating. Now there is an original The Hacker's Dictionary by Guy L Steele that was published in 1983. So far I can't find its text online.
There is a chance it may have had such a slang use in another language. European cars only used chrome very sparingly. I checked German Wiktionary and found nothing but maybe a big official dictionary has something.
I suspect the information from the Jargon File and (New) Hacker's Dictionary is both right and wrong. I believe the step about "automotive slang" was a brand new folk etymology from the writer's imagination trying to fill the gap without evidence or experience of car enthusiast terminology. But the part about wargaming sounds very plausible to me. I know when I was getting into computers almost before colour and hi-res that many of the older guys were into wargames. I never took an interest.
My personal feeling, which I haven't found conclusive evidence of yet, is that I first saw the modern GUI sense when Netscape went open source after losing the Browser War to Internet Explorer. Whether it was in the original source, or whether it was tied to their XUL interface markup language during the Mozilla days before the Firefox branch became the main browser, I do not know. It is definitely used in XUL markup though:
Oh and MDN, the Mozilla Developer Network has a whole article on this sense of chrome, which links to its glossary entry, which in turn links to this definition on a "Nielsen Norman Group" site, whose author states they don't know the origin but speculates that it's: "likely a visual analogy with the use of metal chrome on big American cars during the 1950s".
I wasted a few hours trying to establish whether it was previously used in X11, Mosaic, Motif, etc but failed to track it down. Even when you try to search specific dates, both Google's browser and results with wrong dates are far too common. Perhaps it came from one of the graphical environments of one of the proprietary Unix variants?