There is a common misconception that money is not value, because it can not be eaten etc. But, all value is one individual's preference for one thing over another, which includes money. Therefore, money is value just like other things.
I agree that value is subjective, means utility and can be compared on the basis of an individual's preference of one thing over another. But I don't agree that money can be judged that simple. One of the fundamentals of the subjective value theory that you make reference to here is the notion of diminishing marginal utility. Money apparently doesn't conform to this notion, which can be explained only by the assumption that money doesn't possess value per se. It is just an intermediary for (or, rather, into) things that can be bought with it...
That's a good question
The diminishing marginal utility still applies to fiat money, but because fiat money have universal acceptance (In fact only domestically), can buy you anything, then the diminishing marginal utility will apply to all the things combined together, thus very difficult to observe for average household. You need much larger amount to see the effect
Imagine that you have 100 trillion dollar, adding another 100 trillion dollar is almost useless to you, because it does not buy you any more things than 100 trillion dollar can do. In fact for 50 trillion dollars you might already run out of the things that you can buy in the whole world (Many things are not for sale), this is the problem that central banks are facing right now
You could just say that the money has no value by itself, and the money an individual has represents the amount of goods that he can have. Thereby you can exclude money out of the equation completely and would lose almost nothing from the picture. If we consider value as utility, then money has some utility to us by and of itself, since it gives us the possibility to choose what to buy (and what not) as well as postpone buying altogether. Whether this utility abides by the law of diminishing marginal utility (i.e. diminishes with the amount of money an individual has) remains to be seen...
Personally, I don't think that it does, therefore there is room for doubt about the nature of this utility