its kind of a term of art and applies more to the state than the police specifically - although the police are obviously the primary tool of doling out legalized violence and have in many ways usurped the power to determine what qualifies as legal.

> For the same reasons, "monopoly" does not mean that only the government may use physical force, but that the state is that human community that successfully claims for itself to be the only source of legitimacy for all physical coercion or adjudication of coercion. For example, the law might permit individuals to use force in defense of one's self or property, but this right derives from the state's authority.

https://en.wikipedia.org/wiki/Monopoly_on_violence

One difference is that regular citizens have to prove that they were acting in self defense while, in practice, police are almost always given the benefit of the doubt.

When using that, it's not talking about self defense. The government has a monopoly on the use of violence in the sense that if you owe them money or some other obligation, they may go through paperwork, collections, all the normal resources. At some point, though, they will send people with guns and physically collect you. Don't pay the IRS for long enough and the government will use violence against you, generally in the form of locking you up against your will.

Ticketmaster’s monopoly on venue event ticketing doesn’t mean it’s illegal to sell tickets if you’re not Ticketmaster, and doesn’t require laws the specifically say Ticketmaster gains the monopoly

No, it's not true at all. It's just somebody who heard the phrase applied to the state and then forgot the context and decided it sounded smart to transpose that to the police. Even as applied to the state, it's an oversimplification meant to make a point rather than describe reality.