Clarification: Bitcoin or any currency, asset or collectable doesn't literally "store value". Value is just a mental operation. 
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Yes! A lot of people use AE to justify that all value is subjective. But value has both an objective and a subjective part. The thing is that economics only deals with the subjective part. The exception to that is money, and that's partly why monetary economics is so hard.
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Concrete example of objective value?
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Writing a text exactly about that! Coming soon!
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It imagine you are choosing between two types of gasoline to fuel your car. There’s an objective measure of use-value (octanes per volume, or something like that). You can objectively say that one is X better than the other. But you should talk to a chemical engineer.
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I think Menger used the term "utility" to designate this objective aspect, might be more appropriate term to make a clear distinction from the subjective concept of "value".
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I certainly have to dig deeper, but I think Menger used the word utility in a very different meaning from the one we do nowadays. The term utility was used by him as a property of any good (like air has utility). But only economic goods can have use-value.
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Maybe the distinction between objective/subjective use-value/exchange-value came later, from B-Bawerk. But both Mises and Rothbard used it broadly. I remember getting lost in Theory of Money and Credit by how much Mises mentioned these concepts and assumed the reader knew them.
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Not familiar enough with Bohm Bawerk's work, but I'm certain Menger made this distinction (though maybe not explicitly enough). I remember he called the disparity between this objective and subjective valuation "imaginary value", which I find pretty interesting.
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I'm not so sure all intersubjective things like markets have to rely on object knowledge. Perhaps to the extent they do is because they're two sides of the same thing, but markets often run on information asym or imperfect, certainly the boom and all other "animal spirits"...
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IMO "intersubjective" refers to explicit & implicit agreements and long term traditions that "live" in the memetic/cultural space whether or if they meet objective needs of the phenotype, as defined by its genetics, is where they rely on objective reality, if not they disappear
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