Yes, it could be bone, though I can't quite see cellular structure. But if there is cellular structure, it could alternatively be wood. Reptiles, dinosaurs and plants are known from Port Mulgrave (amongst other fossils).
Thanks for those.
I do not see the sort of cellular structure I would expect from bone or wood. But there is also the possibility that your specimen is a cast of the original - in which case, internal cellular structure would be absent (it could be preserved at the surface).
I hope somebody else comes up with an answer for you; a lot of identification has to do with raw experience (and I am more of an invertrbrate chap).
If not, I suggest you show it to the folks at the Whitby Museum (local knowledge is such a useful thing)
Your find is interesting but I don't think it's bone unfortunatly, although there looks to be an area of fossilisation near the base of your specimen.
I hope you do not mind as I have attached a couple of images of Plesiosaur bones also found at Port Mulgrave on the low water mark to help with id.