dB(A) is a weighting. It’s not a reference and it’s not units. I think some of the confusion here comes from people not actually understanding units.

A-weighting describes how different frequencies are summed up. It’s like saying “RMS”. RMS is not units, A-weighting is not units. You can apply A weighting to voltage, digital signals, or audio. They all have different units but can all be A-weighted.

You could invent a new unit for A-weighted audio, but you would need several.