I think I've seen discussions about this before, and obviously the USSR produced art because we still see statues of Lenin today. But how does this translate in modern times with the instance of obscure art or other modern art? Often the purpose of that art is to explicitly go against societal norms for aesthetics.
Sure, anything made for exchange is a commodity. I was incorrect in saying this art is not a commodity if it’s made for exchange. But Marx is observing and analyzing generalized commodity production under capitalism (The wealth of those societies in which the capitalist mode of production prevails, presents itself as “an immense accumulation of commodities”). How commodities are produced under capitalism is unique. He is looking at commodity production while the means of production are privately owned, where wage labor predominates, and commodities are produced for sale in a market. Under these conditions, any given commodity is produced at a sufficient scale that there is a socially necessary amount of labor that must be used to produce it, and in the production of this commodity, since labor as a commodity is used to produce it, surplus value also is generated. When talking about someone making their own art with their own tools for exchange, while it is a commodity, the rules of value and surplus value don’t apply in my opinion, so I would still argue what OP is asking about is outside the bounds of Marx’s analysis.
The Law of Value is for commodities, not just commodities produced through the standard capitalist formula of M-C...P...C'-M'. Artisinal production of commodities, be they art or otherwise, still confront the sphere of circulation as commodities the same way commodities produced through traditional industrial production do, even if their production circuit is distince.
Artisinal production still follows in the general value calculation, it's still produced for exchange. No surplus value is extracted from an artist that produces commodities to exchange for their necessities, but the value is still regulated around the socially necessary labor time. The Law of Value is concerned with commodity production, capital, etc, not just capitalism.
What's unique about capitalist production is the dominance of capital, ie the ability for a capitalist to buy commodities with a sum of money, have these work together in production, then sell the produced commodities of higher value for a greater sum of money, ie M-C...P...C'-M'.
An artist that buys their own materials and tools isn't also buying their wage labor. There's no surplus value extracted from them, the form a part of the small handicraftsman. The law of value still applies to how their produced commodities confront the market. They practice M-C, do their own P, then convert the new C' into M', but usually cannot move on to reproduction on an expanded scale except perhaps in the guild format. They get their labor-power "for free." The C that they purchase just takes the form of mp, they supply their own L. As compared with other commodities produced artisinally, they will still be regulated around the socially necessary labor time. Rates of profit tend to be high for the petite bourgeoisie, but gross profits tend to be low due to the artisinal scale.