The ability to see invisible can be possessed by any creature, including the PC. The PC can obtain this ability by wearing items that grant it, such as the ancient mummy wrapping or a ring of see invisible. It is also possible to throw a potion of visibility at a creature to temporarily remove its ability to become or remain invisible.
Creatures cannot obtain the ability, but many possess it by nature, and all creatures will be able to detect the invisible PC for the rest of the game if they were attacked by them just once either in melee or using ranged weapons. The more prominent creatures that see invisible include the orb guardians and many other "boss" NPCs. Invisibilty remains a tremendous asset in the elemental temples since the many creatures they surround themselves with for protection typically do not see invisible.
The ability to see invisible is separate from the ability to see in Darkness, which tends to be somewhat rarer.
While most of the most powerful creatures in the game see invisible, removing an otherwise risk-free way to defeat them using only spells (including wands), there are notable exceptions, the most notable perhaps being the Ancient Karmic Wyrm. While Sharad-Waador is the only one naturally appearing in the game and deep in the CoC, the PC can wish for additional ancient karmic wyrms if so desired at any time. If the PC has access to Invisibility and enough wands or spell charges of Magic Missile (the only spell capable of harming the wyrm), patience is the only thing separating them from an enormous amount of experience points, especially if the PC uses Slow Monster on themselves (which reduces speed, which increases experience gain) beforehand and turns the wyrm itself invisible (which provides another XP boost). This boost can be significant enough to kickstart, for instance, a level twelve PC all the way to level 50.
Note that if the PC can see invisible, (s)he will not get an increased XP boost from killing an invisible opponent that (s)he would otherwise get. If attempting the tactic of XP grinding with the ancient karmic wyrm detailed above, the player would have to remove any means of seeing invisible from the PC.