It is half true, on an application form you need to list a reason for a rifle, but not a shotgun. The list of reasons for a rifle are the same in the USA as the UK with the exception of self defence. The main legislation is the Firearms Act 1968
http://www.legislation.gov.uk/ukpga/1968/27
Our Unwritten Constitution, nicely summarised by Wikipedia
"Unlike many other nations, the UK has no single constitutional document. This is sometimes expressed by stating that it has an uncodified or "unwritten" constitution.[2] Much of the British constitution is embodied in written documents, within statutes, court judgments and treaties. The constitution has other unwritten sources, including parliamentary constitutional conventions (as laid out in Erskine May) and royal prerogatives."
It is pretty much a case of pass a law in the UK and it becomes part of our constitution.
"Acts of Parliament are among the most important sources of the constitution. According to the traditional view, Parliament has the ability to legislate however it wishes on any subject it wishes."
We have the right to guns as much as you do in the USA. Its is in our Unwritten constitution as per the Firearms Act 1968. The Founding Fathers did not like the idea of an Unwritten Constitution, so they wrote one.