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The current (electricity) produced by a fuel cell scales with the size of the reaction area where the reactants, the electrode, and the electrolyte meet. In other words, doubling a fuel cell’s area approximately doubles the amount of current produced.

I think there's proper meaning in dictionary. What's this meaning here?

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I don't see the difference of the meaning of the verb "scale" between engineering and everyday life.

When you buy some water, the money you spend scales with the amount of water. Assume a bottle of water sells for $1, 10 bottles cost you $10, 55 bottles cost you $55.

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It was hard to find this use of "scale" (intransitive) in a dictionary. I did find a similar meaning in Wiktionary
Wiktionary "scale" (verb, intransitive)

3: (intransitive, computing) To tolerate significant increases in throughput or other potentially limiting factors. for example, That architecture won't scale to real-world environments.

The use you quoted seems to extend that use, from computing or industrial processes, to electrical engineering. It's quite understandable, but "scales with" seems to mean no more than "is proportional to".

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I can't really find a good definition of the engineering sense in a dictionary but if we start from this sense of scale:

a ratio of size in a map, model, drawing, or plan. "a one-fifth scale model of a seven-storey building"

So I can buy a 1:100 scale model ship or a 1:50 000 scale map. If I buy a scale model I expect it to accurate, apart from the size.

In engineering and science, as your original text says, it means "proportional to" or "approximately (linearly) proportional to". For example, if I say: "The heat loss in a fusion reactor scales as the surface area of the plasma but the heat production scales as the volume" that tells you that large fusion reactors are better.

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