Most lenses from the major manufacturers have a proprietary connector that the hood uses to attach to the front of the lens. While you can purchase hoods that attach to filter threads, that doesn't guarantee the hood will work properly with a lens it can screw onto. This is because:

I read that the focal length plays a role...

It's even more involved than that: focal length plus sensor size play a role since ultimately it is about the angle of view. This distinction is most often applied to telephoto lenses, which yield a fairly narrow angle of view. There are a few cases of lenses that have two different hoods available: one for the wider angle of view yielded with a larger (typically FF sensor) and another for the narrower angle of view yielded with a smaller (typically APS-C) sensor.

If you want to have some fun with this concept, try printing and constructing a paper/cardboard lens hood for your lens from lenshood.net which will be for use on APS-C cameras and another hood for the same lens from lenshood.co.uk that will be the correct size for that lens when mounted on a Full Frame camera. Then compare the two hoods. If you use the hood designed for the crop body on the FF camera, you will see vignetting as a result of the hood blocking some of the light that would wind up in the part of the image circle cast by the lens that falls on a FF sensor but that does not fall on an APS-C sensor.

If you want to go the economical route, you're probably better off buying a third party hood made to fit to a particular lens, such as this one for under $10 meant to replace the OEM ET-87 from Canon that runs about $60. There are also a variety of other third party ET-87 copies available at various prices and quality levels. Other hoods for other lenses have a similar variety of options out there as well.

It's also not just focal length that pays a role in hood design. If the front of the lens rotates during focusing or zooming, a circular hood should be used that is as short as the shortest corner of a tulip-shaped hood for a lens with the same angle of view and front element size and position.