Money (the medium of exchange) is an asset. But the demand for money is different from the demand for any other asset. Here is an (inadequate) parable that tries to explain why it is different.
A dozen kids are sitting in a circle around a campfire. They have cold hands. The parent has a bucket of hot yams. (They are magic yams that stay hot forever). Each kid has 2 hands and wants to hold exactly 2 yams. Trying to hold more than 2 hot yams would burn the kid. So a kid holding 3 yams would immediately pass one on to his neighbour, who will only accept it if he has an empty hand, or sees there's another kid with an empty hand.
With 12 kids times 2 hands each there is a demand to hold 24 hot yams. The parent hands out 24 yams and the kids pass them along until each kid is holding 2 yams. If the parent tried to hand out a 25th yam, no kid would accept it. Because any kid who accepted it would be unable to pass it on to another kid who wanted it. The stock of hot yams held by the kids is determined by whichever is less -- the kids' demand or the parental supply.
Now let's change the story from yams to potatoes. Unlike yams, a hot potato will only burn you if you hold it longer than one minute. You can hold as many hot potatoes as you like, providing you don't hold any potato for longer than one minute. So the kids hold each potato for exactly one minute, then pass it on to the next kid in the circle.
What is the demand for hot potatoes? How many potatoes should the parent hand out? 12? 24? 36? 48? This question doesn't (yet) have an answer. The parent can hand out as many hot potatoes as she likes, and the kids will accept them, hold each potato for one minute, then pass it on to the next kid. The stock of potatoes held by the kids is determined by the parent's supply, and not by the kids' demand.
How would we need to change the story to make the demand for potatoes determinate?
Suppose the kids also supply and demand backscratches. You can only give a backscratch with your right hand, to the kid on the right. So you can only get a backscratch from the kid on your left. Suppose the rule is that the price of a one-minute backscratch is one potato (I'm assuming sticky prices). Potatoes circulate clockwise, and backscratches circulate counter-clockwise.
Let's start with a recession. There are only 6 potatoes in circulation, so 6 kids are producing backscratches, and 6 kids are consuming backscratches, at any point in time. You might say that the demand for potatoes is also 6, because that is the number of potatoes needed to pay for the backscratches that are being produced and consumed. But if the parent hands out a seventh potato, it will be willingly accepted, and be passed on once a minute, so now 7 kids are producing, and 7 kids consuming, a backscratch, at any point in time. If the parent hands out 12 potatoes, the recession ends, because each kid is now fully employed giving backscratches.
The stock of potatoes demanded equals the number of backscratches produced per minute. Because velocity of circulation is 1 per minute. But if the parent increases the supply of potatoes, the number of backscratches increases, and the demand for potatoes increases in proportion. The supply of potatoes creates its own demand.
What happens if the parent tries to hand out more than 12 hot potatoes? The kids will still accept them, because they can always pass them on to the next kid. But they won't get a backscratch in return. Because all the kids are already fully employed. If it weren't for the campfire taboo on changing prices, we would see the price of backscratches begin to rise.
Should we say that the kids demand a stock of potatoes equal to one minute's worth of nominal income from selling backscratches? Wouldn't it be better to say that the kids have a desired velocity of circulation of one per minute?
Money, as the medium of exchange, is like the hot potatoes in this parable. All other assets are like the hot yams.
Perhaps we should abolish "the demand for money", and speak about "desired velocity" instead. Or, better still, talk about the desired degree of synchronisation between payments and receipts.