Does lapis lazuli really dye stuff like in mine craft?

I don't think it was ever used to dye fabric, but it was used as a component in a pigment called Ultramarine, which was very expensive and in high demand by painters.
Lapis Lazuli is a rock, and to make a dye out of it, you'd need to either make it liquid (not happening) or dissolve it in something (possible but not without altering it in undesirable ways). But for a pigment, like a paint, you can 'suspend' a powdered form in something like oil.

Yes