The state's monopoly, qua Max Weber, is on the claim to the legitimate use of violence. That is, the right and legitimacy of that right, is restricted to the state, or an entity acting in the effective capacity of a state, whatever it happens to call itself.
Absent this, one of three conditions exist:
1. There is no monopoly. In which case violence is widespread, and there is no state.
2. There is no legitimacy. In which case violence is capricious.
3. Some non-state power or agent assumes the monopoly on legitimate violence. In which case it becomes, by definition The State.
The state's claim is to legitimacy. A capricious exercise would be an abrogation of legitimacy
Weber, Max (1978). Roth, Guenther; Wittich, Claus (eds.). Economy and Society. Berkeley: U. California Press. p. 54.
<https://archive.org/details/economysociety00webe/page/54/mod...>
There's an excellent explanation of the common misunderstanding in this episode of the Talking Politics podcast: <https://play.acast.com/s/history-of-ideas/weberonleadership>
The misleading and abbreviated form that's frequently found online seems to have originated with Rothbard in the 1960s, and was further popularised by Nozick in the 1970s. It's now falsely accepted as a truth when in fact it is a gross misrepresentation and obscures the core principles Weber advanced.