If having a dollar means having some thing or other, then no one could have a dollar if there weren't any. But bits of language of the sort "have a ___" often don't call for filling the blank with the name of a thing. If someone has a cold, or an idea or a worry or a lot of work to do, there isn't some thing they're carrying around in their nose or their head or have stored in their office.
Unlike colds and good times, dollars once were bits of stuff and mostly still are -- at least in one sense. But having a dollar has never been simply a matter of having a bit of paper. What makes the piece of paper a dollar is that the person who has it has a certain amount of economic power, so to speak (which, by the way is another example of "having" something that's not a thing.) Money is already a lot more abstract than the creased bits of cash in your wallet let on. And so in the cashless economy you're imagining, that's what having a dollar or a euro or a million such would amount to: having what we're calling, rather crudely, a certain amount of economic power; having the capacity to acquire goods, services, whatnot with a certain value.
And as you in effect point out, we're already quite comfortable with this idea. If you're anything like me, most of your financial transactions these days consist of swiping debit cards, filling in online forms and signing on various dotted lines. Most of my available cash exists only in abstract form. When I get paid, it's by way of an electronic transaction between my university and my bank. I don't even get paper notifications anymore.
Of course, somehow this doesn't stop me from ending up with pointless pocketfuls of pennies, which I'm told are worth more these days as copper than as cash!