-- A wave travels from place to place.
A pendulum just sits there.
-- A wave carries energy. A pendulum has energy,
but doesn't take it anywhere.
-- A wave is a traveling pattern of maximums and minimums
that all exist at the same time but in different places.
A pendulum is a stationary pattern of maximums and minimums
that all exist in the same place but at different times.
(I made this one up just now; I really like it.)
What makes you think that all oscillators must be pendulums,
or must be waves, or must be both, anyway ?
A guitar string is an oscillator that has waves on it,
but no pendulum anywhere nearby.
A car horn is an oscillator that sends waves out
but there's no pendulum anywhere nearby.
An analog wrist watch has an oscillator in it, which is
actually a torsion pendulum, but there are no waves
anywhere nearby.
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Going just one step further with the logic of your question ...
If the logic held water, then we would have to ask:
A horse and a dog are both animals.
So why isn't a horse a dog ?