All, or at least most property rights are monopoly rights anyway. I have a monopoly right over my house, and my car, my bank balance. That's just what ownership means.
Those rights are very flimsy actually. The government can seize your house, your car, and your money anytime. Hardly a monopoly when a third party can break it at will.
That the state which grants you your right can take them away doesn't make them flimsy.
And it's certainly more than "hardly" a monopoly. If the government gives a certain company right to operate on train track infrastructure but denies the same to every other company, then does that first company hardly have a monopoly?
By that standard, nobody has any right to anything. I think it's pretty widely understood that rights range from aspirational descriptions of a just world to widely accepted legal consensus.