No, MOVs have no care. Capacitors don't provide any surge protection, just provide filtration for a power supply on an active logic circuit in a suppressor if present.
Inductors don't provide any significant surge suppression either, though are often present in mid-grade or better suppressors as a noise filter (suppressor has a line filter feature in addition to surge protection) and then the frequency could matter but not so much at a mere 50/60Hz difference... that's not the noise they're trying to filter.
It is unlikely that any of them can't work at both line frequencies but if one were designed to protect very sensitive equipment then one designed for 110VAC could theoretically shunt voltage too soon on a 220VAC line, but you'd know you're buying something unique in that case and most often I suspect they use the exact same design for both voltages. If you have equipment that sensitive it should have its own built in protection.
"Maybe" they choose a little higher voltage MOVs for 220VAC. I'm talking consumer grade which is meant more to deal with lightning surges, not lesser voltage kickback from your fridge or A/C compressor, etc.
Consumer grade primarily depends on MOVs and they'd blow out (too soon) if they were rated too close to line voltage. Granted these days there are consumer, prosumer, pro, industrial, etc - The line blurs but you won't get things like spark gap on a cheap unit (maybe just closely spaced PCB traces meant to act like one, where legal ).