The Vegan Society does do certifications for products which will let you use their logo. I’m not aware of any laws which would stop you from calling a product vegan without that certification, just can’t use their logo for it.
I can’t remember if it was the EU or ISO which was working on a standard to classify a product as vegan in the past couple of years. Though the last news I heard about it was that it was being led by agricultural lobby rather than animal rights activists.
I’m not sure wether that logo is really as expensive as they claim, but there are tons of products out there with kind of selfmade logos or at least a little text which reads “vegan”, “vegan friendly” or “suitable for a vegan diet”. Of course the official logo is the safe one, but as it’s still illegal to make misleading claims I guess one can trust 99% of those labels.