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Take one of those cheap power inverters and try to measure the output voltage with a multimeter. You'll find that only the good ones with DSP (or the old digitals with the really good analog front ends) will measure properly. Not even a classic analog meter will read properly.
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But DSP is NOT a requisite, it's just ONE of many ways. Analog or digital back-end is not relevant. An ordinary needle meter or a digital voltmeter can measure the proper value sought after with the proper front end depending on what you're looking for. If you're looking for RMS value, you use an RMS converter or if you're looking for average, you use a circuitry known as the integrator.
Integration is a mathematical function used to gather the cumulative quantity (collected quantity). The 1 and 4 shown in pictures are called limits of integration or a span of 3. A divider is used to continuously divide the collected sum to produce the average value.
If each shaded square meant a joule and there are 150 squares in there and the units for the x value (the x is the side-to-side axis), expressing it as "watts" means it is divided by 3 so it will read as "50 squares/second" or 50 watts. An analog integrator can provide the value continuously.
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It is possible to do it in analog (as those old Flukes show), but in practice, a modern meter is going to use DSP. It's simply much cheaper to throw in a processor than it is to mass produce a bunch of precision analog circuits.
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It's a mass production, non DIY adaptation friendly, highly mass production friendly, digital bullcrapping and mass China made production optimized setup.