Think of it as a scale, but with the scale being observer dependent. Your clock, and your ruler will define distances, time, as well as the relative motion of others. But, in reality, you're asking if the universe has a way to 'know' what a motion is, and there I think the answer is yes. 'Energy' is how the universe defines 'relative motion' and you gaining 'energy' will be a result of your acceleration, no matter if you define it relative Mars or Alpha Centauri.
So yes, the universe keeps 'count'. But that 'count' is nowhere in a uniform motion, you can't measure it. And to me it must have to do with 'room time geometry' somehow. But macroscopically there is no way I know of that you can define a 'energy' in a (uniform) motion. You can use some relative 'frame of reference', like the cosmic background radiation, or Earth, or measure the blue shift of light from distant stars, but in a black box scenario?
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The question is trickier than it seems. We define energy from 'invariant mass' acting for example relative our Earth, and assuming that motion is relative our definitions of the 'energy' expressed in a collision, then they all should become arbitrarily made as we have no way of defining a absolute rest frame, only a relative. Or maybe its just me :) finding it trickier to define than it should be.
Then again, the universe has one definition of being at rest, uniform motion, and there is also the one where you're at rest relative something else. But I like uniform motion myself :)
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Yep, that's a perfectly acceptable way of defining 'energy' locally, using 'uniform motion' as a state of rest I think. And then imagine yourself at rest relative it.