The right to allocate and charge moorings arises in common law, much as the right to rent out premises does. It's not a legislative issue so whether to allocate a spot and whether to provide it to red, green, blue or even purple (or aubergine) boats is a matter of contract, should the lessor agree to enter into one in the first place.
I'm not going to google it (I don't need to, it's in common law) but there is, in the 68 Transport Act, license for BW to lay on and/or maintain whatever facilities they so choose I'd imagine that was intended to codify the common law aspect.
I must remember to log out...
Love smelly, who likes kittens as well