To ‘feel cold’ you have to have a heat transfer between 2 objects. Take a book and a metal plate at the same temperature. The book doesn’t feel cold(or hot) because when you touch it very little heat is transferred. The metal plate does feel cold because, despite being at the same temperature as the book, metal can hold more heat at a given temperature than the paper book. So heat transfers from your hand to the plate.
Basically different things have a different heat capacity. This is also why steam burns are way worse than boiling water burns, there is more heat.
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