It's often misreported this way in the popular press but your ADIZ is not "your airspace", and establishing an ADIZ is not a claim of sovereignty.

Right, I was just staying consistent with how the person I responded to was using the term.

Is there a legend for that map? I'm not sure if I'm interpreting it correctly or not.. I see the box over Taiwan, but that extends over into the mainland. Do they really scramble jets in cases where Chinese fighters don't even go over the water? Or is that line that goes over the water the start of what they consider to be their airspace?

I don't think they scramble jets on Taiwan, no. But every time a news report comes out about Chinese jets representing escalating aggression towards Taiwan, that is the boundary they are actually talking about.