Is a car red? Some cars are red, some aren't. Some red things are cars, some aren't.
Government-granted monopolies can create property rights, sure. Other property rights (even intangible property rights) exist that are neither government-granted nor monopolies - obvious example, a debt. And a government can grant a monopoly without creating a property right, e.g. by choosing to award all its rail contracts to a single company.
I think the word "red" isn't a fair comparison. I'd use the word "love".
If Pat told Alex, "I love you", Alex would rightly infer that Pat had just expressed romantic interest. If Pat then said, "...like I love my dog," Alex would then rightly be confused (and angry).
In the same way, the term "intellectual property rights" overloads the phrase "property rights". It's a poor choice of words and since "intellectual property rights" is the neologism, it should change. Otherwise, we'll have to come up with a new term for natural "property rights" and teach everyone what that term means.
And debts are not properties, they are stipulations in contracts that can sometimes be transferable. There are rights involved with forming contracts, but they aren't the same as property rights.
AFAICS we just have different priorities in what we want the terminology to do. You're prioritising the impression that the term "intellectual property" creates in a layman, i.e. that it calls to mind an analogy (with tangible property) that is in many ways misleading. I'm prioritising the technical accuracy of it as a legal term per common law definitions of proprietary right.
So I think we'll just have to agree to disagree here, based on our different sets of priorities.
(As for debts, in English law they are choses in action, and so intangible moveables. See e.g. Lord Goff's comments in Lipkin Gorman v Karpnale [1991] 2 AC 548, p. 574 at [A]. So a debt isn't quite a purely contractual right: the difference is between a claim in debt, e.g. an action for the price, and a claim for damages. The former will often allow some kind of proprietary remedy - e.g. a lien over goods if the debt is from a sale of those goods, the ability to trace your title to the debt into other people's pockets (as in Lipkin, which was a purely personal action for unjust enrichment), or in some circumstances even a full-blown constructive trust (e.g. in a Chase Manhattan v Israel-British Bank type situation). The latter never will.
That said, I did perhaps choose a bad example, as the line can be a thin one. A better example of a property right that doesn't give you a monopoly might've been be an easement. (Freehold covenants are another interesting example, as starting in the mid-19th century, they've sortof morphed from purely being personal, contractual rights into being property rights)).
I can agree to disagree, though it's worth noting that intellectual property law is currently under review in courts and legislatures worldwide. It follows that the meaning of "intellectual property" is, among other things, a significant political problem right now.
Is a car red? Some cars are red, some aren't. Some red things are cars, some aren't.
Government-granted monopolies can create property rights, sure. Other property rights (even intangible property rights) exist that are neither government-granted nor monopolies - obvious example, a debt. And a government can grant a monopoly without creating a property right, e.g. by choosing to award all its rail contracts to a single company.