Although I disagree with you here. I like to think of distributed ledger systems like Bitcoin and Monero as a new type of money that has even more features than money of the past. You said yourself it's a store of value (cash stores my value) and it's a private medium of exchange (just like cash). The blockchain can admittedly be used for things that have nothing to do with money like domain names and voting systems.
I guess my question is, is your point of contention with calling Monero money that you fear people will think of it as only money and disregard its other potential uses? Obviously it has some of the properties of what we traditionally think of as money (and possibly more) but I don't know if it's helpful to try and stop people from calling it money. For now it's the best word we have. I also like digital currency (virtual currency is bad because it implies it is fake or ephemeral). Cryptocurrency is also nice but it is not as useful as money or digital currency when explaining these things to laypersons.
Yes - it's primarily due to the limitations we'll place on ourselves if we only think of it as money. I am talking more from a technical perspective than an economic one, since my grounding in economic theory is admittedly shaky, and I know that the association with traditional money is unavoidable (hence the ISO 4217 compliance in the XMR symbol). At the same time, we don't want to limit ourselves and our thinking.
With regards to the "layperson" understanding it: once we start down the road of saying things like "digital cash" the layperson goes "oh, so like PayPal?" Then anything awesome like M-of-N multisig (which Monero supports in the protocol just not in the client yet) is completely foreign, because you keep having to force the analogy. The same problem exists if you use "just like gold and silver" in place of "store of value", because people have this preconceived notion that gold and silver have value because of scarcity and utility (when most of its value is artificial and market driven anyway). You'll also run into difficulty trying to apply the analogy with regards to Monero's untraceable and unlinkable nature.