Many tools/APIs provide ways of selecting elements of specific classes or IDs. There\'s also possible to inspect the raw stylesheets loaded by the browser.
However,
I think the answer from S.B. should be the accepted one at this point but it is not exact. It is mentioned a few times that there will be some rules that may be missed. Faced with that, I decided to use document.querySelectorAll instead of element.matches. The only thing is that you would need some kind of unique identification of elements to compare it to the one you are looking for. In most cases I think that is achievable by setting its id to have a unique value. That's how you can identify the matched element being yours. If you can think of a general way to match the result of document.querySelectorAll to the element you are looking for that would essentially be a complete polyfill of getMatchedCSSRules.
I checked the performance for document.querySelectorAll since it probably is slower than element.matches but in most cases it should not be a problem. I see that it takes about 0.001 milliseconds.
I also found CSSUtilities library that advertises that it can do this but I feel its old and has not been updated in a while. Looking at its source code, it makes me think there may be cases that it misses.