Note that we make a distinction here between what is “technically recyclable” and “actually recycled” (more on that here).
It is easy to recycle plastic wrap in industrial quantities.
Clear “stretch film” is often recycled when it comes in large quantities with high levels of purity (eg no tapes, stickers, or labels). Logistics and shipping companies use stretch film - also known as pallet wrap - to secure boxes on shipping pallets. A lot of this gets recycled. Yay!
However as “post-consumer waste” (the stuff individuals throw into a recycling bin), it is generally not economically feasible to recycle. It is mixed together with other types of plastic, it is in very small quantities, and it is often contaminated with food waste. All this makes it difficult to process.
Clean plastic wrap is welcome in blue bins however it is unlikely to be recycled in practice due to these technical difficulties of sorting and collecting sufficient clean material for sale to recycling companies.
Meaning do not go out of your way to clean this material: if it is dirty, throw it out.
However as long as it is clean, putting it in the blue bin does no harm but understand the chance of it getting recycled remains low.