I disagree that it's even socialism as NYC isn't outlawing or using the emoluments clause to take control of private stores. It isn't ceasing the means of production in any sense.
The Mamdani plan is to put in stores where no stores exist. That's just a city ran store. Something that used to be pretty common in the US.
Stores exist there. They have some foodstuffs even. They have bars on all the window and your money is handed through a rotating tray under a bulletproof glass with tiny holes in it so you can talk to the cashier (who also has a shotgun resting under their cash register).
That is because that is the viable model in those areas to actually run a store without having your staff egregiously injured/assaulted and not have everything not nailed down stolen.
This ends up getting reflected also in higher prices of foods in those areas, to reflect the cost and lower supply of those willing to take those measures. So people will call it a 'food desert' -- not because you can't get food there (though it mostly sucks and is shelf stable stuff) but because nothing there resembling a walk-around middle-class grocery store exists.
Failing to take those measures, or taking the measures and not raising prices enough to cover the costs, will likely end up in the state losing money, that is, they will be forced to seize the capital of private citizens to fund the state's commercial offerings.
I think you mean "eminent domain" – the emoluments clause prohibits government officials from accepting gifts, payments, or titles from foreign states.
Communism only happens after a revolution overthrowing the current government and replacing the entire economy with the state overseeing it. Note the word entire.
Socialism is when you have the public own entire industries. For example, how the oil industry in Norway is owned by the state.
Having the government do something or own a business is neither communism nor socialism. It's simply a state owned thing. It's not, for example, socialist for the government to have a parks department.
A city ran store is not the city owning the means of production. There will still be private stores throughout NYC. The areas where these stores are being targeted are where those private stores have chosen not to deploy.
Interestingly "no government" also appears to be exactly what is happening in the US with what some people might describe as end stage Capitalist society.
