Most wrapping paper can be recycled if it is indeed paper. This also includes Christmas cards, gift bags, and cardboard boxes. One way to test if its paper is to scrunch up the gift wrap in your hand. If it remains balled up, there’s less of a chance of it containing foil which means it can go into the recycling bin.
If you still have the original packaging the gift wrap came, you could check its composition to make sure it’s 100% paper.