I've found a stuck handle or two in 15 years as a jobbing handyman, but that experience wasn't in the USA.
Here, most shafts are either square and tapered (usually) or splined (sometimes).
New taps are rarely seized, it seems to be the ones that have been together for 20 years that give trouble.
The working mechanism part of the tap is replaceable (here!) from a plumbing merchant. I HAVE cut the shaft off rather than stress the tap/faucet. I've also tapped the handle upwards to remove it, depending on what it is made of. If it's metal, it is probably brass, and so easily marked with a hammer.
If the tap has been leaking for a long time, it may be that the seat that the washer tightens down on is damaged and must be recut.
If there is a metal dome over the mechanism (after the handle is removed) this needs to be heavily wrapped in sticky tape so that it is not gouged by the tool used to unscrew it.
Hope this is useful. What Rap said up there at 07:08 is sense.
There are three seals in a tap, an O-ring around the handle shaft, a gasket washer under the mechanism, and the tap-washer. You should also use a special non-poisonous grease on the mechanism threads.