The performance of the analog RMS chips like AD736 or AD8436 is not that great: The accuracy and BW is only good for not too small an amplitude and the frequency range is somewhat limited to the low side (e.g. already no longer really RMS below some 20-40 Hz, depending on the filter).
It does not need that much resolution (e.g. 12-14 bit ) to get comparable or better performance, with different weak points for both systems.
The ADC can be sampling, maybe a little better than those found in µCs, but also a SD ADCs. AFAIk the DMM chip-sets with integrated RMS use SD ADCs - so the same technique as the main ADC for DC, just faster.
The digital energy / power meter chips use something like 16+ bit 100 kHz range SD-ADCs, somewhat similar to the audio ADCs, though with better gain stability. They need some of the resolution to get good accuracy over a large current range (like still 1% accuracy with 0.1% of the FS current). Chances are some of the BW limiting is intentional because of the standards for power metering.
A more normal handheld meter should get way with less than 5 digits for AC and thus less ADC resolution.