What's the name of the fabric that's used for covering speakers or what's the closest fabric with similar characteristics that I can use?
Traditionally, it was called "grille-cloth" and it was made from a coarse-weave of rather thick plastic filaments. It is still available in many places, particularly those vendors who supply people restoring vintage equipment.
Modern speakers tend to use more conventional material similar to (or identical as) common, inexpensive polyester double-knit. It is available in dozens of colors (and patterns). And it is "stretchy" so that it can be stretched over a frame and hold its shape. We sometimes even sew up a complete "sock" that goes all the way around the speaker enclosure (as for something visible hanging from the ceiling, etc.)