Two factors make stay ink stay on glossy paper: ink components and paper coating.
Printer ink includes cyclohexanone to help the ink cling to slippery glossy paper. It also includes butyl urea to slow down paper shrinkage as it dries.
Inkjet papers have a special coating to prevent the ink from being runny. It also prevents the paper from softening. The ink does not settle on the paper directly but on this coating. Inkjet papers are also coated with bright emulsion to give the print a bold sheen.