Some wrapping paper is recyclable whilst other types are not.
The easiest way to tell is to do the ‘scrunch test’. Scrunch the wrapping paper up into a ball. When you release your hand, if the paper remains in a ball then it is recyclable in your green recycling bin as it will be paper based. If, when you open your hand, the ‘paper’ springs back then it’s likely to also have plastic mixed in and will need to go into your black/grey waste bin.
Alternatively you could use reusable gift bags or scarf wrapping which can both be reused many times.