Fein allegedly did invent the Oscillating “multi-tool”, or at least get a patent on the design for use as a tool.
Fein wasn’t by any means just a “woodworking” power tool manufacturer though.
The oscillating tool design seems to have originally been for medical use, and I believe is the type of saw that may be used for cutting open skulls for autopsies, although the medical tools are branded differently, and I think routinely have the blade oriented at a different angle to the tool (like a die grinder, but oscillating).
The oscillating tools are also used in orthopedics for removing casts, and I’ve come across one or more orthopedic suppliers selling Fein units, since they seem to be cheaper than the versions from medical tool suppliers.
Some of the medical branded versions used to be made by Fein, private label, judging by looks, just in a different color and branding.
The Fein Multimaster, and the Fein vacuums seem to be what got Fein noticed in the woodworking trade, since both were widely available from quality woodworking tool suppliers.
The Fein Multimaster was mostly marketed as a detail sander, even though saw blades were available, probably because the original saw blades sort of sucked. (Fein only included a regular, relatively soft steel saw blade with the Multimasters, and the steel was significantly softer than their HSS blades).
The Fein vacuums, were sold because they were incredibly quiet, and were rated for use with asbestos once outfitted with a hepa filter.
The fein plunge router, as I mentioned above, was manufactured in Italy, by a power tool company called Fellisati.
Fellisatti, seems to have been a known power tool manufacturer in Italy, but outside Italy, the tools were mostly made private label for other brands like Skil and Porter Cable.
Most of the Fein tools marketed to woodworkers, were not actually made by Fein, other than the Multimaster, one or two cordless drills, and a couple random orbit sanders.
Fein tried marketing a jigsaw model in the USA, but it was less versatile than many other cheaper options, such as the Festool Jigsaw and top of the line Bosch jigsaw, because the Fein saw designed for cutting thick steel, unlike the other two.
The German Fein manufactured tools were comparatively expensive, especially compared to other “Industrial Brands” sold in North America, and seem mostly to have been marketed to industrial users.
Fein has had tool distribution in the USA from a headquarters in Pennsylvania, going back probably to the 1950s or 1960s.