Mostly because your mouths "wet", a ton of energy is required to heat or cool, water/saliva acts as thermal sink any hot soup or cold popsicles transfer energy only changes a few degrees and absorbs heat readily when changing state (ice/steam). If your mouth or tongue/gum skin were real dry it would be burned or freeze immediately instead. People with cavities, weak enamel, gum disease or Stomatitis are VERY hypersensitive to cold and heat in mouth is painful to them.

Conversely if you ran your hand under hot or cold water before touching something you'd be fine for a bit, is why wet newspaper is used for glassblowing, etc https://m.youtube.com/watch?v=eGqln2ZvTMQ

[removed]

[removed]

Our sense of perception of heat is "hard-wired" to warn us about an environment that's going to suck out our body heat (unpleasant feeling --> avoid when possible), but not to warn us about drinking cold water, as it would have been a bad idea to skip an opportunity to rehydrate when water is available.

When you put your hand or body in cold water, your body's natural response is to try to warm itself up. It does this by sending more blood flow to your skin and by making you shiver. This can make you feel uncomfortable or even cold. On the other hand, when you drink cold water, it can actually help to cool your body down. This can be especially refreshing on a hot day or after you have been exercising. It can also help to quench your thirst. That's why it can feel pleasurable to drink cold water.