“Measured like any other natural disaster” is a bit vague. Some natural disasters have relatively basic repeatable characteristics that can be measured and relate strongly to the effect they have, and so we have designed scales for them. Hurricanes and Earthquakes are the only ones I can think of though, and there’s been a lot of discussion recently of how the Hurricane scale is becoming less useful as other factors than wind speed were responsible for the extent of damage in Houston and in the Carolinas.
Other natural disasters are measured and characterized based on their likelihood, which doesn’t necessarily tell you much about their effect, although there is a correlation between the extent of damage and whether there is a 50-years flood, 100-years flood or 200-years flood.
Arctic ice loss and permafrost erosion lends itself to neither of these approaches, but as already mentioned we can of course quantify it and monitor it. But I’m not sure if that is what the OP intended to ask about.