No particular style of printing.
It's a diecut circle - it slots precut for the chopsticks during diecutting process.
And the text is printed in black
And then debossed.
You might have a sheet like this printed with colour.
And the text printed in the middle
Then the text is pressed into the page by a block of wood/metal, either by hand or by machine.
There's also hand debossing machines and things like that.
It would all depend on your budget/print run sizes.
You can buy debossed circles online - and you could hand print onto them using screen print, or a digital printers and deboss them yourself.
But your image looks professionally done.
Fonts
Sushi - I'd say it's Sublima Semi Bold
Samba - I'd say that's Neona