They do. But not as a tourist. You have to live in the country.
If there is some other option, it's a "grey spot" (to bypass the law).

This is the second time I have heard of foreigners not being able to register vehicles in the UK and I am not sure how they will know as there is no nationality or identity check, just don't tell them you are a foreigner, all you need is a UK address. You will have to tell the insurance company but that is a different matter.

A few thoughts:
Here in Denmark I can register a bike and then register another person as the user of that bike. Maybe thats possible in the UK too.

Maybe you could have your friend register the bike and then you could rent it off him with a contract and all.

You must be joking?
No identity check?
We (SLO) have on line register (no way you can get past that).

Despite being one of the most spied on people in the world in the UK, as far as I know there is no check of where you come from when registering a vehicle in your name, we have no national ID card and there is no place on the registration document for national insurance number, driving licence number or any other identifying number, at least there wasn't when I last bought a bike about two years ago, things may have changed since then.

There is a space on the V5 for a driving license number, but it's optional (and clearly marked as such). But it doesn't matter since they don't bother with identity checks. There isn't even a vague hint of intelligence at the DVLA considering the way my new V5 came through the other day addressed to "Miss Matthew".

Awesome, thanks everyone for the information!