"Monopoly on violence" or "use of force" is the way I've seen it put by political scientists defining what it means for a state to be sovereign (that quality is really high up the list of things that go along with that). It doesn't mean only the state can legally engage in, say, violent self-defense, but it means the state is capable of dictating where and when "legitimate" violence may occur, and backing up those dictates with its own violence, with great effectiveness.

Not that this means a sovereign state may choose to make violent self-defense legal, for example, without any risk to its sovereignty, provided it's still the force deciding whether a given instance was legal and backing up those rulings with threat of or actual violence. It's the unchallenged (un-successfully challenged, at any rate) arbiter of what is and is not permissible violence.

The police are the primary point of contact between the citizenry and the state monopoly on violence when it is most acutely exercised (rather than merely threatened or practiced more softly), so framing this as the police personally having (been invested with) that monopoly, in the context of a meeting on the street between non-police citizens and the police, seems reasonable to me.

A state that finds itself unable to claim or maintain its monopoly on violence is at serious risk of loss of sovereignty, usually to whichever group is having success challenging that claim. Understanding how that balance sits can be a way to insight into the de facto power structure of a governmental system—for example, the de jure sovereign power of a state might be the civil government, but if it effectively operates at the pleasure of a military bureaucracy that has demonstrated the ability to topple the civil government and maintains a tacit threat to do same again in the future, then at best the civil government shares sovereignty with the military bureaucracy, and at worst it's a kind of puppet-state of the actual sovereign power (this, specifically, is why strong standing armies are seen as a risky tool for a state to keep—rather than increasing the power or control of the supposed sovereign, they can easily turn to challenging it very effectively). Turkey's operated this way, at times. Egypt, too. Not that uncommon an arrangement, the de facto sovereignty resting less securely with the de jure rulers than one might suppose.

Looking elsewhere in the thread, reports of police deliberately failing to use their (threat of or actual) force to uphold the law in the districts of politicians who've tried to do something they don't like (reduce their funding)—assuming they get away with it and aren't brought to-heel (apparently they have not been)—is another example of a force-bearing arm of the state threatening the sovereignty of the civil government.

[EDIT] on a more personal note, I'm highly skeptical of the utility of widespread private ownership of small arms as a means of keeping the state in check. I've not seen them to be a significant factor in successful resistance against a state, and especially not against one's own state. For one thing they're plenty likely to be used to enforce tyranny or perpetrate injustice on behalf of the government, officially or unofficially (often the latter—see the post-reconstruction South for a close-to-home example) and for another they seem to be neither necessary nor sufficient for armed insurrection to succeed. The critical factor seems to be overt or covert support by part or all of an actual military, whether foreign or one's own, to provide materiel and direct support (e.g. no-fly zones, factions of the army joining the rebels in a civil war, that sort of thing). I see them as a distraction. I wish the left would stop talking about them because it just drives some people away and they've made little progress on restricting firearm ownership anyway, but I also don't see them as somehow key to preserving liberty, simply because they seem secondary at best in any successful rebellion I'm aware of (except maybe against very weak states with tiny militaries that fail to levy more loyal soldiers before the shooting starts, but that's not, you know, the US, at all).