I don't know if it's better for you than plastic or really anything about its chemical composition, but it's not plastic.
Edit: After reading more, there is an acrylic lacquer that was developed in the 1950s that contains plastics, but outside of that, normal or traditional lacquers are typically biological in origin.
For stuff made by lone craftsmen & small boutique shops, maybe 50/50? Maybe less?
For the lacquer you buy in the stores, you will have to examine the contents to see if you have actual lacquer or synthetic petroleum lacquers.
As far as wood products, if it was made after the early-mid 80's, assume that if it was lacquered it was done so with a synthetic, or if it is some form of manufactured wood.
https://www.chicagotribune.com/news/ct-xpm-1986-11-09-860324...
So I wouldn't assume anything acquired finished by someone else is free of such things, regardless of whatever once-precise names they used to describe the finish.
Even when you try DIY-finish something with an eye towards food-safety it's rather challenging. E.g. the "Boiled Linseed Oil" you find on store shelves isn't actually boiled linseed oil, it's usually been adulterated with petroleum distillates and sometimes toxic "dryers" (cobalt comes to mind) to achieve the results boiling used to serve.
Here in CA the once common Turpentine which is literally distilled pine trees can't even be sold to the public due to VOCs. The stocked replacements are mostly petroleum distillates, labeled as known to cause cancer in the state.
The oil industry has infected everything
It's probably better than plastic, but remember that coral snake venom and puffer fish toxin are 'all natural' and so the concept of all natural is utter and complete bullshit.
I could find images for Indian Brass Container to learn what it is. But I didn't find anything conclusive for Korean steel.