This is a good mechanic and you are applying it well.
Suggestions:
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Add shadows of the other world so it is easier to land jumps
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Can you have a key other than E for switching? Left Shift
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Make the player have the speed of the platform they are standing on summed with their movement speed, perhaps with ray based detection
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Remove friction from the rigid body of the player so you slide down when you rub against platforms
I like where you are going with this and would definitely playtest future versions.