Another clickbait article title. Shame on you BBC. Not fake. "Balsamic vinegar that is not labelled "tradizionale" (traditional) is a blend of wine vinegar and grape must, made from grapes with no particular provenance and aged in wooden or steel barrels for just a few months."
So what would make it fake?
Italy has local standards called, DOP and IGP, if I recall correctly. DOP is the elite stuff. You dont put this on salads. Some varieties are almost gel like. This is the stuff that was sitting in someones attic for ten years. IGP is still good but not as good, and it still has to come from the right region. Costco in the US sells Kirkland balsamic vinegar di Modena, and it is IGP. You can imagine that at Costco scale, it is not sitting in someones attic. This is still excellent, by the way.
Then there is all the rest, that claims to be balsamic vinegar, but was produced to meet high scale, low price points.
The article is simply saying that what you likely buy in the supermarket is unlikely to be DOP and might not even be IGP. Thats not really news.