Metal feels colder than wood at the same temperature because both materials conduct heat at very different rates, not because one is actually colder than the other.
Your body constantly gives off heat to the surfaces it touches. The thermal sensation depends on how much heat a material can draw from your skin per unit of time. Metal is an excellent thermal conductor, meaning it pulls heat away from your body quickly, creating that impression of intense cold. Wood, on the other hand, is a moderate thermal insulator that doesn't transfer heat with the same efficiency.
How heat conduction works
When you touch a metal surface at room temperature, the metal's molecules absorb heat from your skin and quickly distribute it throughout the material. This constant thermal flow keeps your skin giving off heat, so the temperature receptors in your body interpret the sensation as cold.
With wood, the opposite happens. Its molecules don't transfer thermal energy as effectively. When you touch a wooden table, the heat from your skin warms a small surface area of the material, but it doesn't spread any further. That warmed spot acts as a barrier that reduces additional heat flow, so the sensation is one of warmth.
A homemade experiment
To test this, place one thermometer on a wooden board and another on a metal plate in the same room. Both will register the same room temperature. However, when you touch them, you'll notice clear differences because what you're feeling isn't the object's actual temperature, but rather the speed at which that material exchanges heat with you.
This principle also explains why marble feels cold while a similar plastic feels less cold, or why in winter the ceramic surfaces in a bathroom are especially uncomfortable to the touch.
