The parameter in question does NOT have any associated units, it's just a factor or ratio, ie, dimensionless. dB are just a peculiar kind of "unit", because they also express dimensionless quantities, just on a logarithmic scale.
The significance of dB for audio (close to what volume "feels like") is appealing, but for me, it is completely negated by my constantly getting confused by the various conventions to be aware of: Is it in reference to a certain base level? Are we talking about power ratios or about voltage ratios? And believe me, I am reasonably capable on the math/physics side of these things.