Road tax was something that existed in the UK briefly at the beginning of the 20th century, but was abolished in 1937. This was partly because it might give people driving motor vehicles a misplaced and dangerous sense of entitlement on the public highway.
Public roads are free for everyone to use, and are paid for from general taxation. There is a tax, called vehicle excise duty (VED), that is payable by people who want to bring a motor vehicle with them when they do use the roads. This is payable by everyone who wants to bring a motor vehicle on the road, irrespective of whether they “are a cyclist”. It is an excise duty much like the excise duty on alcohol, which also does not confer any special rights on the person paying it.
For many motor vehicles, the rate of VED is zero. This is the VED reminder for my own car:
Yes, I pay the same amount of VED for my bicycles as I do for my car. If you don’t want to pay VED when you use the road, then change your car; or better still use a bicycle instead.