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I'm not a lawyer, but I'm willing to have some fun with this idea.
A cursory review of the relevant California Government Code section 420 -- blaze it! -- provides a description of the California state flag, and also a picture of it. Or it would in the print version of the code. While there doesn't appear to be a specific bit of law which authorizes the state to retain the copyright on the flag, there is case law which disallows the state from retaining copyright for "government documents", with exceptions which wouldn't apply here. So it's reasonable to assume that California doesn't have the copyright on its state flag, with it likely being in the public domain.
This would suggest that Minnesota could indeed use the flag to mean something else, the same way anyone can with public domain material. Now, if this occurs outside of California, that state could not enforce any sort of rules pertaining to how the flag is used. Even within the state, California's authority to control how public domain material -- or more broadly, any material at all -- is circumscribed by the First Amendment in any case. The exception would be for those agencies and subdivisions of the state itself, which it can and does control. See Gov Code section 435, which disallows cities from having confusingly similar flags. The other exception would be uses of the flag which perpetuate fraud or some other related crime, since then it's not the speech being punished but the conduct, which happens to involve a flag-related expression. But neither of these really speak to the flag being used by another sovereign entity within the state.
Supposing for a second -- and this is where we're really departing from reality -- the several states had embassies at each other's state capitals, but without the equivalent protections afforded by the Vienna Convention on Diplomatic Relationships. And by that, I mean each state buys land in other states, without creating sovereignty issues, owning that land as any other individual or corporation could. In such a case, if the Minnesota Embassy in Sacramento were to fly the flag of California as its own, what could California do? If they drafted a law like section 435 that applies to individuals, the First Amendment would present a barrier. If the law applies to out-of-state entities, it might run against the Dormant Commerce Clause, in a very broad interpretation of interstate commerce. If they apply it to all sovereign entities operating within the state -- which would include the Minnesota Embassy, since the State of Minnesota owns it -- then the thorny question of state sovereign immunity in state court would arise.
In a California state court, would the State of Minnesota have sovereign immunity? If instead of Minnesota, it were a foreign country like Scotland, the answer would be a resounding yes. But here is a state vs state issue. The proper venue would be a court with original jurisdiction over states, and there's only one of those: the US Supreme Court.
As to what the state of California would assert as a cause of action? I suppose they could raise a criminal violation of their freshly-drafted law, with the risk of devolving into whether a US State has its own rights of free speech, which other states must respect. Alternatively, they could raise an action in equity, such as a tort (MN's use of the flag is costing CA somehow) or defamation (MN's use of the flag asserts falsehoods about CA).
At this point, we're deep into legal fanfiction and it's time to stop haha. Needless to say, I think the situation in real life would be messy if it were to happen.