Any idea how I can create this effect in Illustrator or what it would be called?

 Thanks tromboniator! Looked into Widthscribe a little bit and looks pretty legit. Was there a tutorial you learned this technique from or just from your knowledge of the software?

I would call it 'cross-hatching'. Because of the curves in the lines, that makes it more difficult to achieve.

You could try using one of the cross-hatch patterns that comes with Illustrator, In the Swatches panel, choose Open Swatch Library > Patterns > Basic Graphics > Basic Graphics Lines, and pick something like Grid 1 Pica Lines. Fill a large rectangle with the pattern. Draw the shape of the glass. Select both and choose Object > Envelope Distort > Make with Top Object. 

It will be close to your sample image, but not exactly like it.

Puppet Warp would be a possibility, but not easy to control for precise lines.

This is what Vectoraster can do:

I have version 6. And I don't use it often enough to tell how to achieve the crossed lines. Maybe newer versions can do that on their own. Otherwise I guess I would just apply it 2x to the same image with different settings.

The closest I can come is with Photoshop, but that does not give the curved crossed lines effect.

 Convert to grayscale, then to bitmap with halftone screen dotshape Cross.

Hi, Eurofinsmarketing

I think it's a kind of line screen.
If it is a straight line, it is possible in Photoshop.
If you do it with Illustrator, is it a halftone pattern effect? But it is quite different from the sample.

There seems to be vectoraster7 though it is only Mac.

https://www.lostminds.com/vectoraster7/

I have never used it.