Sorry but this is just plain wrong. ARM ISA is not software it is an instruction spec consisting of opcodes and instruction definitions, regs spec etc.
I think this is where you and many others are confused. Allow me to put on a computer scientist's hat for a second:
www.cs.umd.edu
An instruction set is the abstract architecture of a computer. But that's not quite what it is. What it is is a set of "specifications" that a physical computer system organization as well as the compiler of any high-level programming language have to adhere to. So in that sense, that makes it a set of "requirements" that your hardware and compiler have to meet.
"Software" in the strictest sense of the word, an organization of logic to achieve a certain purpose, is not necessarily adherent to the idea of an "instruction set". For instance, the logic I put together to control a cash register is "software". The translation layer (the compiler) is not necessarily a part of this "software". It's a part of the computing pipeline, or just another layer to get my "software" to work with a certain computer with a certain architecture.
Similarly, I think you are understanding "hardware" in this sense as the "organization of physical parts to perform computing tasks as defined by the software". So yes, in that case, I do agree with you that your computer or "hardware" is not strictly "the" abstract architecture. Your processor can meet the standards as defined in the instruction set, but it obviously can do so much more.
But... in software, there's a similar concept of an "interface". If your organization of logic, a "class" per se in some languages, is implementing an interface, then the convention is: it is what that "interface" is defining. So in that sense, if my software was written with the intentions to be adherent to a set of requirements as defined by an instruction set, then... it might well be what that instruction set is defining. This is is the current industry standard. And it applies to both software and hardware.
Let's bring that back to the current discussion: the ARM ISA is basically like the interface. It enforces certain constraints (requirements or specifications) that your processor has to meet. So your processor, by industry standard, is an ARM processor. And yes, that also means that, by industry standards, if your processor can meet both ARM and x86 specifications, then it's both ARM and x86.
So let's just say you are not wrong, but the industry enforces its way. Just like how ARM enforces constraints on Apple Silicon.
P.S.: that's also why I have to go back and recompile my software for them to work on ARM64. And again... I do understand what you are trying to say and I do agree with you. But that's not how the industry works. My software was previously x86, and now it needs adjustments to be "adherent" to the ARM compiler... and now it's both an x86 and ARM64 software. But hell, I didn't write even a single line of x86 or ARM64-specific instruction.