Yes, there is an industry standard for determining whether a mattress is firm or plush. The firmness of a mattress is usually determined by its indentation load deflection (ILD) rating, which measures the force required to compress the mattress material by a certain amount. (This is not a mattress industry scale it's a foam fabricator's scale)
The ILD rating typically ranges from 12 to 50, with higher numbers indicating a firmer mattress. A mattress with an ILD rating of 12 to 20 is considered very soft, 20 to 25 is soft, 25 to 30 is medium-soft, 30 to 35 is medium-firm, 35 to 40 is firm, and 40 to 50 is very firm. (true)
In addition to the ILD rating, the thickness of the comfort layers on top of the mattress and the materials used can also contribute to the overall feel of the mattress. Generally, a mattress with a thicker comfort layer made of softer materials will feel more plush, (always if both characteristics are combined) while a mattress with a thinner comfort layer made of firmer materials will feel more firm. (thinner firmness layers will never be as firm as thicker firmness layers)
It's worth noting that different manufacturers may use different terminology to describe the firmness of their mattresses, so it's important to consider the ILD rating and other factors when comparing mattresses from different brands. (All that is actually important is how a mattress feels for you and nobody else)