Silk Printing: Ink + screen (mesh stencil) = screen printing, support color printing.
Hot Stamping: Heating up a colored foil and melting it on the bottle. Gold or Sliver are popular.
Decal: When the logo has too many colors, you can apply decals. Decal is a kind of substrate on which text and patterns can be printed, and then transferred to the surface of the bottle.
Label: Custom a waterproof sticker to paste on bottle, multicolor possible.
Electroplating: Use the principle of electrolysis to spread the metal layer on the bottle.