What style of illustration is this? It appears to be like an old style but I haven't been able to find a name for it. Despite the text on the image, it's not "vintage".
It's a style of line art which simulates relief prints made from a woodcut or linocut - where a piece of wood or linoleum was carved/gouged out with carving tools, then the surface inked, and a print made from it.
You can read more about it here: https://en.wikipedia.org/wiki/Woodcut
These methods of printing were used for illustrations/prints before the advent of photography or photographic etching techniques for printing. It was all done manually and was a highly skilled trade. Of course, there are still artists who create such prints in the same way even today.