I’m wondering what you would call the style/technique of the picture above? It looks similar to linocut, and is halftone but neither of those terms are specific enough.
I’d love some advice on how to apply such an effect to a photo in Illustrator or Photoshop as well (can’t google it yet because I’m not sure what it’s called)
It’s a style meant to emulate old engravings (look at the portraits on american paper money for an example). Traditionally it was done by an engraver, cutting into the plates.
The example above appears to be done via a PhotoShop filter. There’s several on the market that can do that. It’s often a form of a halftoning filter.